Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
997
Algebraic Geometry Open Problems Proceedings of the Conference Held in Ravello,.May 31 -June 5, 1982
Edited by C. Ciliberto, E Ghione, and E Orecchia
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Editors
Ciro Ciliberto Istituto di Matematica "R. Caccioppoli", Universita di Napoli Via Mezzocannone 8, 80100 Napoli, Italy Franco Ghione Dipartimento di Matematica, Universita di Roma II Tot Vergata, 00100 Roma, Italy Ferruccio Orecchia Istituto di Matematica "R. Caccioppoli", Universit& di Napoli Via Mezzocannone 8, 80100 Napoli, Italy
AMS Subject Classifications (1980): 14-06 ISBN 3-54042320-2 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-38?-12320-2 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are macle for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Introduction
The
Conference
held
in
Ravello
1982
. This
ring
the
to
all
lume -
. We
the
, the " Ente
wed
the took
finally
tion
only
the
theme and
thank
" Consiglio
the
ce We
also
or
in
use
as
of in
" Banco
the .
thank
all
of
the
week
the
papers is
Algebraic : May
lectures
grown paper
up
to
from
. We all
31
st-
and
n ° 15
Conference
particular
Geometry
June
talks
5
th
given
discussions
, which are
" was
duamong
however
extremely
contributors
,
fits
grateful
of
this
vo-
of
Na-
:
di
delle
Napoli
Provinciale of
in
the
exception
Nazionale
place
the
most
well
The
participants
pies -
with
as
problems
) during
contains
Conference
well
" Open
( Salerno
volume
participants very
on
per
wonderful
those
developement
Ricerche
" il
, for Turismo
" Villa
persons of
the
who
"
, the
their di
financial
Salerno
Rufolo
"
helped
Franco
"
support for
, where
either
Conference
Ciro
University
in
.
ciliberto Ghione
Ferruccio
Orecchia
having the
the
; allo-
Conferen-
organiza-
A.
Conte
: "
Enriques
S.
Greco
: " Remarks
C.
Peskine
W.
Fulton
E.
Arbarello
: " : "
List
of
threefolds
"
on
the
curves
: " A
few
Mumford
: "
F.
Catanese
S.
L.
A.
Beauville
C.
De
E.
Sernesi
J.
Harris
The
: " : "
Concini
The
: " On
a
about
given of
moduli
on
the
Mehta
: "
Vector
C.
Ceresa
: " Remarks
of the
P.
Maroscia der
: " Geer
to on
Hilbert
: "
The
G.
Van Gupta
: "Schubert
R.
Smith
: "
M.
Beltrametti
H.
Hulek
D.
Eisenbud
: " Rational
E.
Stagnaro
: "
P.
Craighiero
: "
The
: " The
on
Conic
normal
"
and
locus
bundle
on of
curves
general
" type
given
"
ones
varieties
"
certain of
linear curves
series
: "
Cubic
of a
varieties
a
and
their
with
set
modular of
Prym non
"
finite
Siegel
space
Enriques
surfaces
theoretic
"
"
map
points
n
in
threefold
representation
" theory
in
curves
surfaces
"
"
"
complete
surfaces A3 C
"
"
rational
cusps
of
3 Constructing
"
"
geometry
geometry the
bundles
of
touching
equivalence
of
of
curves
varieties for
plane
"
function
and
of
unirational
projective
geometry
irreducible "
Talks
curves
calculus
branch
varieties
projective
of
of
genus
space
uniqueness
algebraic
The
R.
surfaces
,,
surfaces
symmetric
bundles
restriction
and
of of
List
V.
algebraic ~3
variety
moduli
rational
problem in
the
space
complete
Problems
in
degree
enumeration
Problems
: " On
: "
of
the
of
curves
"
geometry
: " On
Kleiman
of
things
curves D.
singularities
Classification Nodal
Lectures
whose
intersection
from curves "
quintics are
in
set-
P
11
List
A. A l b a n o ( University rello ( R o m a I ) , D°
of
Participants
of Torino ) , E. A m b r o g i o ( Torino ) , E. Arezzo ( Genova ) , M.G. Ascensi ( Brandeis
Arba) ,
F. B a l d a s s a r r i ( Padova ) , E. B a l l i c o ( P i s a ) , U. B a r t o c c i ( Perug i a ) , A. B e a u v i l l e ( Ec. P o l y t e c h n i q u e , P a r i s ) , G. B e c c a r i ( Torin o ) , B. B e l l a c c i n i ( Siena ) , M. B e l t r a m e t t i ( Genova ) , J.F. Boutot ( Strasbourg ) , M. B r u n d u ( Genova ) , M. C a n d i l e r a ( Padova ), F. C a t a n e s e ( P i s a ) , M. C a v a l i e r e ( Genova ) , G. C e r e s a ( Torino), L. C h i a n t i n i ( Torino ) , S. C h i a r u t t i n i ( Padova ) , Ciro Ciliberto ( Napoli ) , A. C o l l i n o ( Torino ) , A. C o n t e ( Torino ) , M. C o n t e s sa ( R o m a I ) , M. C o r n a l b a ( Pavia ) , P.C. Cralghiero ( Padova ) , V. C r i s t a n t e ( Padova ) , C. C u m i n o ( Torino ) , C. D e C o n c i n i ( Roma II ) , A. D e l C e n t i n a ( Firenze ) , P. D e V i t o (Napoli ) ,Ao D i S a n t e ( Napoli ) , D. E i s e n b u d ( Brandeis ) , P. E l l i a ( Nice ) , D. E p e m a ( Leiden ) , G. F a l t i n g s ( Wuppertal ) , M. F i o r e n t i n i ( Ferrara ) , M° F o r m i s a n o ( Napoli ) , P. F r a n c i a ( Genova ) , W. F u l t o n ( Brown), S. G a b e l l i ( R o m a I ) , R. G a t t a z z o ( Padova ) , F. G h i o n e ( R o m a II), A. G i m i g l i a n o ( Firenze ) , S° G r e c o ( Torino ) , Guerra ( Perugia ), R. G u p t a ( M.I.T° ) , J. H a r r i s ( Brown ) , D. H u s e m o l l e r , K. H u l e k ( Erlangen ) , M. I d ~ ( B o l o g n a ) , S. K l e i m a n ( M.I.T. ) , W. K l e i n ert ( East Berlin ) , Kooler ( Brandeis ) , A. L a n t e r i ( Milano ) , E. L i M a r z i ( Messina ) , R. M a g g i o n i ( Catania ) , M. M a n a r e s i ( Bologna ) , M°G. Marinari ( Genova ) , P. M a r o s c i a ( R o m a I ) , G. M a r tens ( Erlangen ) , C. M a r t i n e n g o ( Genova ) , C. M a s s a z a ( Siena ) , L. M a z z i ( Torino ) , V. M e h t a ( Bombay ) , I. M o r r i s o n , D. M u m f o r d ( Harvard ) , , G° N i e s i ( Genova ) , F. O d e t t i ( Genova ) , P. O l i verio ( P i s a ) , A. O n e t o ( Genova ) , F. O r e c c h i a ( Napoli ) , M. Palleschi ( Parma ) , G. P a x i a ( Catania ) , C. P e d r i n i ( Genova ) , U. P e r s s o n ( Stockolm ) , C. P e s k i n e ( Oslo ) , L. P i c c o Botta ( Tor i n o ) , A. R a g u s a ( Catania ) , L. R a m e l l a ( Genova ) , S. R e c i l l a s ( Firenze ) , L. R o b b i a n o ( Genova ) , N. R o d i n 8 ( Firenze ) , M. R o g gero ( Genova ) , D° R o m a g n o l i ( Torino ) , M.E. Rossi ( Genova ) , G. Sacchiero ( Ferrara ) , P. S a l m o n ( Genova ) , F.O. Sehreyer ( Bran, d e l s ) , E. S e r n e s i ( Roma I ) , M.E. Serpico ( Genova ) , J. S h a h ( Northeastern ) , R. S m i t h ( Georgia ) , E. S t a g n a r o ( Padova ) , R. Strano ( Catania ) , E. S t r i c k l a n d ( Roma I ) F. S u l l i v a n ( Padov a ) , G. T a m o n e ( Genova ) , G. T e d e s c h i ( Torino ) , C. T r a v e r s o ( P i s a ) , C. T u r r i n i ( Milano ) , Ughi ( Perugia ) , G. V a l l a ( Genova), Go V a n d e G e e r ( Amsterdam ) , , B. V a n G e e m e n ( Utrecht ) , G. V e c chio ( Catania ) , L. V e r d i ( Firenze ) , A. V e r r a ( Torino ) , G.E. Welters ( Barcelona )
TABLE
E.
BALLICO On
A.
and
and
bundles
of
a proof
of
examples
of
general
Torelli's
of
16
....................
34
........................
90
...........................
113
surfaces
type
theorem
algebraic
EISENBUD
and
J.
the
Brill
threefolds
surfaces
whose
hyperplane
............................
124
............................
131
HARRIS:
- Noether
theorem
FALTINGS: of
Arakelov's
intersection
product
.............
138
FULTON: nodal
FULTON,
curves S.
the
...........................................
KLEIMAN
and
enumeration
R. of
146
MACPHERSON: contacts
.........................
156
GHIONE: probleme
fibres GRECO On
the
GRUSON
du
type
vectoriels and
A.
Brill-Noether
Postulation
les
........................................
of
singularities
and
pour
197
VISTOLI:
construction
assigned L.
non-rational
Enriques
Un
S.
....................
unirationelles
FRANCIA:
are
About F.
on
et
sections
On W.
P.
surfaces
Properties W.
i
CONTE:
On G.
......................
CILIBERTO:
Two
D.
curves
CATANESE:
On A.
projective
rationelles
BELTRAMETTI
Moduli C.
of
BEAUVILLE:
Conic F.
CONTENTS
ELLIA:
degeneration
Vari~t~s M.
Ph.
OF
C.
rational
surfaces
with
....................................
210
PESKINE:
des
courbes
gauches
...........................
218
Vlll
K.
R.
P.
HULEK: Projective
geometry
LAZARSFELD
and
Linkage
general
of
problems
points V.B.
in
MEHTA
and
MORRISON The
A.
group
the
RAO: curves
of
large
results
on
finite
degree
....................
267
sets
of
U.
of
290
in
characteristic
p
.......................
315
PERSSON:
sections
on
a rational
elliptic
surface
.........
321
Kodaira
dimension
variety
of
the
Siegel
..............................................
348
ORECCHIA:
ROBBIANO Some
Hilbert
and
curves
G. in
functions
of
Cohen-Macaulay
varieties
.... 3 7 6
VALLA: p3
are
set-theoretic
complete
intersections
surfaces
quintics
in
.... 391
STAGNARO: Constructing
G.
228
RAMANATHAN:
bun~es
Generalized
E.
.......................
.................................................
and
modular
L.
curves
MUMFORD: On
F.
and
pn
Homogeneous
D.
elliptic
MAROSCIA: Some
I.
P.
of
VAN Prym
DER
Enriques
from
p3 K
...........
400
GEER:
surfaces
and
a Siegel
modular
threefold
.................
404
ON DEGENERATION OF PROJECTIVE CURVES
by
In
this
of
projective
p n
a smooth
C
at
only
a scheme that
study
curves.
one
In
A
C
point
and
BALLICO
Ph.
ELLIA
problem
of
embedded
typical
example
is
as
of
degree
(
L
fiber
the
and
the
Zt
and
smooth
case
to
P nT
of
we
deformations
follows
a line
tangent
Z is
affirmative
d
not
a subscheme
generic
?
we
curve
T
the
C U L
paper
E.
L
C
: given
) does
' flat
in
intersecting there
over
and
the
special
will
say
that
T
exists such
fiber
C U L
is is
smoothable. If
C U L
denotes ves
the
of
genus
smoothable, of
genus
one On
the
non
other
a stronger schemes be
space of
able
and
hand
and
of
the
n)
some main
, a line no
from in
by
d
reducible idea
of
our
every
embedded that
In
~k
)
C
curves paper
,
of is
in
the
as
we
C
only
is prove
small
same
sub-
way
a linear
closure
~k Prd(C,
at
0C(I)
fact
~k
the on
is
curve
C
if
degeneration
~n
cur-
deformation.
In
111.5)
which
curve
a smooth
intersecting
method
of
this
d+1 ; n)
nonsingular
not
d+1 ; n)
in
Prd(C,
projections
way
exists
L
the
Thm
Z(g, of
smooth
Z(g,
degree
in set
the
there
has
(see
is the
. By
8
is
denote
general
d IV.2)
constructing
of
it of
L
follows
d+1;
curve
to d e s c r i b e Indeed
by
Z(g,
~n
(see
C U
it
then pn
degree
C U L
result
set
Hilb
degree
then
of
in
that
a given of
the
smoothable
example
and
such
special
C
g for
9
point
is
closure
in
Then ~k
follows
)
let sub-
Hilb we
are
(see
II)
: take
a
~k
curve
Y
ne
of
H
ting
in
Y
, a line
. Consider
D
and the special
C U L
(see
just variations
intersecting
D
: the generic
fiber
1.1)
Y
at one point and an hyperpla-
the flat family of curves
from the points of
Pr(Y, H) curve
p n+1
~n+|
:
Y
. This is our starting
111.4
and Theorem
111.5
N
about generic
projections
ved by Tannenbaum with a different
X
and
Y
of
if they are nonsingular
Tx X
Tx Y
If
C
~n
---~ Tx ~ n
we denote by
ty and X
write
of F
C an
in
H
OX- module
:
L(D)
:= L
x
d
we put D
~
(see [ Ha,2]
that Theorem 111.5.
closed
field
K
(resp of maximal
in
~n in
Prd(C; :
F
and
H
4.3.4)
had been pro-
with
ch(K)
Hilb H
H)
rank)
and
. We denote by
at
: .
of the set of general
is irreducible.
X
= O .
is a linear subspace of
:= H O m o x ( F , 0 X)
a divisor on O(D)
curves
and if the natural map
the closure
. Note that
is a smooth curve,
to solve Hartshor-
( [ T2 ] ) .
is surj ective
Prd(C ; H)
§ II
are said to be (quasi)-transversal at
is a smooth curve of degree
jections
If
method
~n
x £ X n Y ~
) will be a reducible
of
of elllptic
We work over an algebraically
Two subschemes
D
point and all our results are
are largely used in [B - E ] . Fi-
As this paper was finished we learned
NOTATIONS
obtained by projec-
on this theme.
Proposition
NOTE
H
fiber will be a smooth curve in
(over the point
nally we plane in a future paper to use the results ne's conjecture
in
L ~X
If
, h°(F)
X
:= dim H°(X, F).
a line bundle on the
pro-
is a varie-
canonical
X
we
sheaf on X.
0x I
.
THE METHOD In this section we prove
truct
embedded
deformations
we will get our results the typical
Prop.
of curves
1.1
which is our main tool to cons-
of type
from little variations
situation we will
consider.
C U L of Prop.
. In the next sections 1.1.
Let us describe
In of
degree H
such
~n+1
(n
d
genus
and
that
:
> 2)
x
take
~
a non
g
, a point
H
. We
degenerate x
denote
of
C
by
~
smooth
curve
and
hyperplane
an
C
: pn+1
.... ~
assume
Y
H
n
the
projection
from
x
. Put
Y
:= w
(C)
. We
is
non-
X
singular
(of
degree
which
the
image
is
d-1 of
) and
x
we
under
denote ~
by
(i.e.
y
the
:
{y}
point
= T
x
In
this
1.1.
situation
at
y
PROOF to
(see
points
of
I)
D
choose
D the
\ {x}
such
genus
over
the
flatness,
cause
every
deed
~
from
exact
the
pa(Y
U
L)
with
the
the
:
, therefore fiber
Y
o
and, line.
set
from
curves,
X CH
degenerate
we
[ Ha]
of
is since This ~ the
it
theoretic
is
( [Ha]
the
degree is
OL
fiber
HD
is
(see
Fig
Tx C --+
OFnL
, II
9.8).
preserved
precisely
x
follows
of
degree
X
tangent ~
number of
of
L
at
D \{x} can
X
Y U
line
from the
, II.9.9)
subscheme
--* Oy U L --* OY
the
C
curves
(
, not
distinct
a finite
point
{y}
x
t of
H D \ {x}
a
at
necessary
x
Y
=
X
of
Y
P r d ( L ; H)
irreducible
to
~ L
only
non
flat
over
projection
:
is
if
is
double
the
sequence
schematic
X
contains
has
from
C
in
a point
projections
a unique
contains
assumption = g
of
intersecting is
C in
smooth
family
H L
a family
restriction
also
fiber
arises
the by
fiber it
the
exists
fiber
in
Y U
L
yields
are
. This
line
then
Since
X
there
the
a
disregarding
, whose
Obviously
and
be
consider
of
X
intersecting
{x}
that,
g
D that
line
. This
fibers
claim
by
L
intersecting
D \
Furthermore flat
a
. Now
by
points, and
be
, and
Fig
parametrized
d
Let
D
C
Y
H)
:
quasi-transversally,
Let
tangent
We
have
PROPOSITION
only
y
we
of
C ~
L I),
. Now ---~ o
that Y
U L
agrees
bein-
4
L
For
further
applications
Iowing
lemma
1.2.
LEMMA
g
and
then
L
for
is v e r y PROOF
Let
a very every
We
have
to
which
LV(S+T-D))
L
is n o n
we
will
need
also
the
fol-
nonsingularcurve
on
X
D
on
that
every
If X
L
the
of
genus
is n o n
line
special
bundle
L(D)
show is
for
equivalent,
we
special
h
o
by
S,T
C
X
: h°(L(D-S-T))
Riemann-Roch,
LV (-D))
(*)
have
h°(~X ~
(~X~
L v)
= 0
.But L v)
to L
being
= h°(~X ~
and
(*)
: very
LV (S+T)).
follows
im-
g
We
and
bundle
divisor
way,
DEGENERATIONS
fact
a complete
= h°(~X ~
Since
In
be line
effective
in a s i m i l a r
mediately
X
ample
ample,
II
I.I .
too.
= h°(L(D))-2 h°(~X ~
Prop.
:
ample
.
of
continue Prop.
Prop.
II.2
OF our
II.1
is
PROJECTIONS study
is just
of
obtained
reducible from
a combination
a
eIements repeated
of
I.I
use
and
in
P r d ( C ; H) of
Prop.
II.1.
1.1
II.0.
PRELIMINARIES
Let d,n
special IV,
X
be a c o m p l e t e
be i n t e g e r s very
6.1)
Define
ample
, and L
:=
As L
of
~
d-g-n
in
and
support
and
PROOF
: Di ~
a curve singular
Suppose
' pn+~
points ~;~(p ~n+i ~ ~_gl in
.
H°(L)
+
x
the point
x
such
that
locus
we have
the ,
(Aj)
to the
does
. Arguing
+'''+P'x-I)(X)
tangent
gives
PI +'''+ Pd-n-g :
... U D d , g _ n to
~O~(X)
, I _< i _< d-g-n
. Then
' P (HO(~)v))
... U A ' d _ g _ n
i+I
spanned x
~
with
same
constructed
by the d i v i s o r
at
U DI U
not
( [Ha ]
Y
U ~'i+1U
where
X
a non
and non special. Thus
:= ~ ( X ) lines
Prd(~L(X) as
on
bundle
(1.2)
and let
: P1,..,Pd_g_n
, we may consider
= {~(Pi)}
in
the p l a n e
i+1 ~ j ~ d - g - n Y" = ~ ( P I
Y
distinct
X
. Since the divisor
Consider
~ ~(p1+...+pi)(P
~(p1+...+p.)(X) 1
]pn
H°(L)
+p )(Pj) "'" i corresponds
given
line
is very ample
~(X) Y
The
g
Consider
g+n in
~0=~ : X ~
~
d-g-n
= ~(p1+...+pi)(X)
Prd(~L(X)
points
: X C.4b~ d-g
: H°(~)
D i' s are
exists
: L
~L
PROPOSITION the
satisfying
there
Y'
]pn
of genus
d > g+n
of d e g r e e
distinct
is non special,
induces an inclusion ]pn C ]pd-g II.1.
;~
curve
and
+ "" + P d _ g _ n ) X
defines an embedding
where
n > 3
line b u n d l e
take
~(PI
an e m b e d d i n g
nonsingular
satisfying
are
general
< j < d-g-n --i n c l u s i o n of Pi+1
i) by not
and
1.1
U A~'z U ... U 4"d_g_n
through
H°(~(PI
consider
contain
as in Prop.
lines
the
and w h e r e
+ "'" + P d - g - n
. Now D
in
the
+'''+P'))z . Denote
a line
tangent
any of the we get is in
that
through
D
to 4! 3
by
'
Prd(~L(X)
, ~
that
~2(X)
where
the
the
(H°(~
U ~I ~l s
points
Next ve
U
~(Pi
such
. After is
in
d-g-n
steps
Prd(~L(X),
r are
m
general
we
have
(H°(~)v))
lines
through
)
a variable
that
= D1
... U ~ d - g - n
, , 1 _< i ~< d - g - n
consider
D (u o)
(PI+...+Pi_l)V))
U
D
line
parametrized
by
a smooth
cur-
:
vu¢
u
:
and
for
DCu)
n~cx)
general
u
= {~(p~))
in
U
:
D(u)
n
U
is
flat
~. l
=
2
morphism
torsion.
first u
part
. A
o
case
if
the at
II.3.
~(X))
.
so
of
It D
D,} 3
D1
(D 1 U
~
borhood
of
y
(Ho( ~ )v)
The
proof
deformed (P2 ) Now
we
Prd(~L(X) quite curve.
happen
.
As
D2
then
Y
U
E
into
not
meeting
give ,~n
different
other )
but
occurs
for
an
yields
that
Y
the
is
dimensional D1
from DI U
and
II.l D~
the
fiber
has
the
over
will
have
we
II.l
y
not
be
the
embedded
situation
of
a curve
it by
proposition.
will this
a point
is
)
: this
hypothesis in
the
Y
Y of
because
' ~n
the
~
, then example
XE(Y)
a three
follows
~L(X)
(XE(y)
containing
in
Prd(
intersect
(H°(~)v)) in
L
in
same
intersect
i
D
process
Under
and
D 2)
(X),
is
the
this
may
{D. n 1
U
fiber
proof use
lines
Prd(~L
~
x
PROPOSITION that
\
the
REMARK
points
Y
of
repeated
II.2.
over
(U
Its; g e n e r i c
have
assume
:
more-
in
in
first
linear
infinitesimal
subspace
neigh
of
D2 )
because where
D~
D 1U is
Dc U a line
~E(Y)
can
be
through
DI examples this
: connected
of
time
reducible
the
chains
elments
configuration of
lines
in of
linked
the to
a
lines smooth
is
Let
Lo
degree union
of
exists such
that an
with o ordering
U
L°
T <
11.4.
a nonsingular
:
i
U
...,
every
order (I)
Y
T
such
for
every
s}
and
(2)
of
~
L°
that
of
length
{O,...,s-1}
In
the
of
with
I
,
L.
II.l
with
get
jecting
from
one
the
get
> i
and
the
of
i ~
r
the there ,
1 ~ k ~ s
the
lines
a function
in
Y
with
for
L
is
o
a reduced
with
a compati-
L
o
in
at
most
one
of
desired
we
x
bamboo
A~
of
is
assume
= ~(p1) (see
D
ni
Y
in
its -=
a bamboo
lenght
. Then
(X) U
point
connected
take
be
~(j)
i ~
number we
Y
simplicity
~(p1) the
II.O
Let
a branch
i ~
For we
j
a maximal
PROPOSITION
PROOF
if
is
11.5.
~
I < i < s
endowed
intersects
is
situation
,
T
lines
i
bamboo
= d-g-n
(Pi )
type s
a branch
deg(D)
~(X)
connected
,
of
be
is
is
for
order
of
transversally
the of
connected
~n which
:
i >
L. [~ L. = (~ 3 z
A branch The
quasi
is
Y
{L.}
Y
i
--
joint
,
a compatible
A bamboo
union
Since
in
Y
in
curve
lines.
U Lk
...
DEFINITION
connected
lines
called
curve
connected
distinct the
is
{I,
for
irreducible
a reduced
s of
L|
ordering
Let T(i)
be
. Consider
L
an
Such
ble
to
d-s
of
r
(Pl) 2)
of
to
I
and , m
and
in
Y
(H°(~)v))
.
d
,P
for at
nI = 2
From
(H°(~(P1))v).
arguing
.
J_
~(X)
Prd(~L(X)
=
lines
P-,
degree
linked in
Prd(~L(X)
Fig.
union lines. r ~ n. i=1
as
in
Then pro II.l
/ llI.
FURTHER
DEGENERATIONS
In
this
section
point
of
view
: is
smoothable C case in
we
prove
~n+~
H(d,
We
non
g;
g
curve
is C U of
we
also L
genus
don't
. This
think
~3 9
curves
result
(see that
the
However
which
is
and d e g r e e
8
not
in
smoothable
is in
a curve enables
us
subscheme
n
of
with
degree
(III.4)
L
, that
this C' to of d is
2 ] )
condition we
C U
Indeed
closed in
[ B-E;
IV
different
condition
This
together
also
type
projecting
(small) of
a
, of
special.
(III.1)
some
scheme
I ]
necessary.
in
by
points
from
sufficient
obtained
its
[ B-E,
a
~n
non
of
in
problem in
is
be
(III.5) in
that
Oc(I)
can
curves
our Y
C
Bilbert
Finally
curve
and
) , the
used
special
one
curve
(III.2)
reducible n
genus
widely
show
that
consider
given
degenerate
from
describe
and
?
is
we
a
O
give
(I) being non c an example of a
with
C
a
non
plane
III.I. of
LEMMA
.
Let
g
and
degree
C
. Then
g
and
genus
point
of
of
genus
is
the
x
as
C
be d
there
of
smooth
non
, a point
from
the
point
of
in
Take
curve
y y
curve
special.
a nonsingular
(d+1) Y
nondegenerate
~C(I)
exists
degree
projection
a
with
Y
Y
a
in
such
, this
~n x
~n+1
that
latter
C
having
image.
PROOF
The
that
there
genus
g
curve
C'
proof
exist and
is
an
degree
in
C'
is
2/
C
Then
" reversing
is
of
the
N
(d+1)
IP N
I/
divided
integer
genus
image
two
> n
, a curve
, a point g
projection
the
into
of
and
of
~
from
under
First C"
of
degree
C"
C'
parts.
we
prove
in
C"
IP N+I of
, a nonsingular
d
such
the
point
a projection
that ~
of
: of
IP N+I
]pN
into
lp n
the
thesis i)
an
of
Let
X
embedding
" the
the
lemma
be
an
with
centers
abstract ~
of
the
projections
we
will
obtain
.
(X)
model
= C
of
. We
the
put
curve L
and
:=
~
~
~
: X ¢----P n (I)
. Let
~n S
be
the
Z
in
for plane
S
sends
x E
{s E
We
L'
ces
S/s(x) :
is
N
of
of
w'
•
h
P E
P
H o (X, the
where
is : X ~ on
very
=
S
ample
. We
~
(S*)
it
~ (P)
bundle
therefore
it
(H°(X, L')*) • Denote L ,) , H o (X, L)
have
by
and
of
~
(Z)'s in
~
a hyper
~n
kernel
defines
an
em-
L)*)
satisfies
defined
the
contained
with
-
is
by
not
X
projection E
~
forms
being
H~(X, |
generated C
. Then
L
P(
L)
. Since
L)
line
•
H°(X,
linear
. Now
: X
ample : X
be )
set
:= h ° ( X ,
sections
~(E*
n+1
the
where
L'
~n
of (I))
in
= O}
a very ~
~
dimension
~ L
put
Let ter
X
L(P)
ding
space
H°(~ n , has
bedding
L':=
subvector
~(U) by
N
> n
in the
Now
= x
By
~
exact
define
Lemma
gives
1.2.
an
embed-
W* , U *
(S*)
from
sequence
:
the
spa-
the
cen-
10
o
Let
~"
be
(K)
the
where
-
E
projection K
is
put
~
:=
The
first
(a)
(c)
~" (C")
ii)
that
have
~
by
center
D
~n
y
o p,,
~
C"
= 0
by
p'
p'
and Y
is
COROLLARY 0C(I)
non
point
special and
:=
(2)
~L,(X)
show
:
the
from
(I)
point
we
have
L
to
the
equali-
L(P)
We
is
the
the
isomorphic
C
and quasi
be let
and of
into
~
P
(W/D)
rational
maps
Y
to from
~
center
a
line
of
W we
~n+1
the
into ~"
and
and
has
degree m
y
curve
in
intersecting
then
W/D
from
°
ge-
that
into
~'
and
L
be
K*
the
degenerate
D
(W/D)
C"
C"
such
:= p " ( C " )
a non
transversally
(h-1(E*))
space
E
image
of
~
because
subvector
put
projection
(2)
points
onto
~(W)
. T he
and
of
a
projection
Let
x
map
be
equal.
is the
point
~,, o ~ L'
choose
of
~(K')
defined
C"
from
number
K'
and
Then
one
may
. Let
center
the
: 0
if w e
isomorphically
the
(1)
from
U ,
sequences
projection
are
ly
ble.
we
D
.
definition =
a finite
the
the
with at
in
(U)
0
p
p"
while
111.2.
exact
by
natural
at
~
projection
~L
the
. Thus maps
:= p " ( ~ )
(d+l)
the only
(W)
from
p'
because
at
h
(D)
Denote
C'
because
C" P
such
=
the
and
Look
nerates
c
.
sequence
~L(X)
=
o
evaluation
h
follows
follow ~L
in
exact
W
:=
(b)
the
intersects
. ~
= ~, true
of
is
S
(W)
the
lemma
~'(c')
~"
is
kernel
the
(c)
~
*
O
~
by
C'
(b)
et :
,
of
(a)
ties
K
~L,(P) part
of
given
0 We
~
C U
L
~n C on-
is
smootha
11
111.3.
DEFINITION
of
genus
of
the
g of
set
put
III.4.
PROPOSITION
PROOF
w,
Consider :=
B
is
closed
X
diagonal
der
the
U(X,
the
smooth
f
d;
u(x,
d; .
X ~
be
the
Z(X•
]pn
n)
very
we
n)
x
~n
to
X
.
X
its
x
~
of
%~here
R
image
]pn
is
x
IP n
to
X
given in
and
B
D
. Consiat
X
).
parametriz~ng . Then
open is
the
IP (H)
D)
"IFn
in
morphism
f
on
(evaluation
Hilb
the
of
F
in
(B
x
I]
follows
ip n
x
~
]pn
dominant
and it
dense
projection
in
U I
by
]]~n
isomorphic a
x -
line
of
projection
h-1(A)
Hi]b
ample U
(n+|).~,
and
X
of
points
:=
irreducible.
defined
open
its
T
the
H
be
is
X
> 2g
x X
diagonal
d
d;
TI
on d
put
]P (H)
T
of
induces
bundle Since
closure
: B k set
fact
is
subset
: B
the
(x
In
n)
the
]"I
and
of d e g r e e
for
\
h
the
)
map
d;
that
Let
:
be
~somorphic
open
dense
and
)
denote x
curves
in
REMARK
X
n)
irreducible -I : ~ (R) • ~'
curves
•
of
natural
Let
Z
Hilb
d
follows
of
]!p (H)
map
Let
(here
the
is
natural
in
a
Poincar4
T
O
:=
Z(X,
factor.
bundle
= B
it
the
subset
Then
evaluation. x X x
a vector
6
>
form
second
, x)/h(x)
the
X
be
the
the
factor.
Consider
F
on
P
{ [ h]
first
by
let
on
closure
degree
curve"
nonslngular
the
as
of
d
assumption d
a complete
n)
28-1)
For
the
. Now
d;
curves
Hax(8+n,
degree
projection
that
F
:-
From of
picd(x) the
6
be
X
Z(X,
nonslngular
We a l s o
bundles
Let
. We d e f i n e
Z(X,d;n)
: subset
of
smooth
exactly
U
From the proof
it follows
that
if
d > 6
sure of the set of non degenerate n o n s i n g u l a r
then
Z(X, d; n)
curves of degree
d
is the clo
isomorphic
to
X III.5.
.THEOREM
Let
D. 1 Suppose
Then
there
•
I < - -
.
Let
i
<
m
I/
C U
2/
C
exists
s
be
, be
D I U is
a
C
a
curve
distinct
.-.
U
D
nonsingular
curve
Y
in
s
in
Z(X,
lines
is
n)
with
d
in
C
U
D s)
contained
> 6 .
connected
along Z(X,
not
d;
C
d+s;
n n)
(D I
U
with
... same
support
12
and
singular
locus
PROOF
We
For (of
s =
course
ly
in
I
induction
:
we
with
one
D I
use
point
assumption
may
and
non
quasi
C
the
on
C
is
nonsingular,
special).
If
transversally
in m a n y
regular
S
s
assume
OC(I)
intersects
I U...UD
CUD
as
points.
part
U
D I
C
degenerate C
then
it
is
111.2.
x°
be
in
DI 0
Let
of
non
inetrsects
contains
x
on-
Suppose C
. By
. Consider O
a
line C
in
{x} no
L(x) x
parametrized
with
. The
torsion.
same
occurs
Now
note
such
that
m(i)
to
L ( x o)
morphlsm
= DI
generic
for
the
the
following
and U
... U
number
in
n
U
is x
fact
:
, intersecting
=
{x}
flat
in
for
because
Z(X,
generic
it
d+1;
has
n)
so
the
o
Dj
there is
lines
U
c is
a curve
over
Ds\
of
x
L(x)
L
fiber
fiber
DI U
the
a point
(U x C)
Its
C U be
by
exists
j
connected.
yoi~ n e e d
to
,
I < j
Indeed,
go
from
< s
,
define
D.
to
C
,
1
pick is
j
such
that
connected.
Now
C U
DI U
the
existence
of
and
singular
locus
If trized
"
•. U
D
intersects
..- U (ii)
6+3
morphic
. We
connected
The
Y'
in
DI U
Di(or U
)
It
may
then
I
C U
(or
Ds
C
and
C U
may
DI U
)
...
U
Ds~
Dj
assume
inductive
hypothesis
Z(X,d+s-1;n) ... U
with
same
shows support
Ds_ I
consider
conclude
From
X C
= @ be
is
,
that
11.2,
111.5 any
and
may
happen
(see
, there to
Dk ~ lines
II .5)
maximum
s >
as
a variable
line
parame-
above.
•
REMARKS
(i)
and
as
D.
1
is
a curve
by
C U DI U
of
is
s-1
Ds
III.6.
d ~
m(j) suppose
it
curve
C U are
2 ~
. We
in
k ~
3
is
follows
(Di)1
found
Y
different
from
II.3). that
in
Z(X,
DI U
D2 U
D3
lines
such
that
don't
Prd( ~ L(X) ' ~d-~
know
if
d;
n)
,
C
is
DI ~
Dk
where
such
(compare
iso-
~
configuration with
11.1,
13
IV
(CONTRE)-EXAMPLES We g i v e
are
not
examples
smoothahle.
IV.1.
PROPOSITION
d > 4
and
Then
let
C U L
of
First
curves
of
recall
the
Let
L
be
is n o t
C
From
Castelnuovo's
it
that
there
follows
Now
we
IV.2.
give
8
a similar
and
point
that
Let
subschemes of H(d, g)s
H(d,
~3
following
a smooth
in
~3
result
plane
intersecting
on
smooth
C
[ T]
example
with
exists
which
:
curve
at
is
of
only
degree
one
point.
denote
of degree
d
of
degree
plane
a smooth
space
curves
d+1
curve
curve
C
intersecting
, genus
C
in C
p3 at
of
only
smoothable.
the
Hilbert
secheme of one dimensional
and arithemtic genus
be the open subscheme of
of
)
L
not
genus
a non
, a line
C U L
the
curve
(see
9
g)
bound
no
There
genus
such
PROOF
are
d > 4
PROPOSITION
degree one
for
C U L
smoothable.
PROOF
(d-1)(d-2)/2
be
a line
type
H(d; g)
g
. As usual let
consisting of smooth connected
curves. A necessary condition for every curve H(d, g)s
,
L
a line intersecting
dim(H(d, g)s + 3
<
dim H(d+|, g)s
Y C
of the type
Y = C U L ,
C
in
at one point, to be smoothable is
, since the choice of
L
is a three
parameters choice. So to conclude we have just to prove the following claim :
(*)
PROOF OF (~)
I
(e)
dim H(8, 9) s
=
33
(8)
dim H(9, 9) s
=
36
(*) this is well known : one easily proves that every smooth curve
:
14
C
of genus
9
, degree
8
is a complete
intersection
F2.F 4
and
(~)
follows. (B) First
Let
Y
be any smooth curve
we o b s e r v e
sheaf of
Y
integral
in
that
~3
~
Then from the
H°(~ 3 , Iy(3))-
and by Riemann-Roch, On t h e
other
a complete
solution
curve
0 ~
:
~ _>
3 (3))
H°(Y,
_>
1
2
because
H°(~ 3 , 0 7
the
ideal
has no
Oy(3))
otherwise
because
3 (4))
. We c o n c l u d e
. Since every
pg 430)
0
--
that
intersection
Y
would
of its genus.
for
N°(Y, Y
be
Now
Y'
Y
Oy(4))
can be
F3.F 4
such curve
the same occurs
R(-3) e 2 F3,F 4
R(-5) ~ 3
[ ELLING 36
denotes
:
is impossible
Y' b.y a complete
and from the r e s o l u t i o n o f C(Y) :
By
Iy
linked
. We have
a
:
is projectively
. Furthermore
to
normal
from the re-
of the cone
C(Y')
sion
<
(
9
:
P a (Y') = 0 Ex I
which
, degree
9 = a+b = (a-1)(b-1)
H°(~ 3 , 0
h°(~ 3 , Iy(3))
F3.F ~
9
= 0
sequence
h°(~ 3 , ~(3))
h°(P 3 , Iy(4))
Cohen-Macaulay
( [ ELLING ]
:
exact
=
H ° ( ~ 3 , Iy(4)) :
= 3
hand
sequence
0 ~
deg (Y~
we get
intersection
from the exact
we o b t a i n
h°(~ 3 , Iy(2))
), this is just because
solution.
0
:
of genus
]
thm.
~ 2
=
R(-2) ~ 3 ~
we get,
R(-4) @ 3
R
as mapping
0 ( JELLING ] loc.cit) cone the resolution of
(1) R(-3)
we conclude
that
. C(y')
R H(9,
9) s
~
= C (Y)
--
0
is smooth of dimen g
15
B I B L I O G R A P H Y
[B-E,
I ]
[ Ha,
2 ]
BALLICO,
E. - ELLIA, Ph. in
~3
HARTSHORNE,
R :
[ Ha]
ELLINGSRUD,
G.
:
238 ,
229-280
of rank 2 on
~ 3 ,,
(1978
" Sur le schema de Hilbert des vari~t~s
Ann. E.N.S.
4 e s~rie t. 8 fasc. 4 (1975) 423-431
[Hi ]
HIRSCHOWITZ,
IT]
TANNENBAUM,
R.
A.
A.
curves TANNENBAUM,
A.
~e
:
:
" Graduate (1977)
" Sur la postulation
g~n~rique
146
" On the geometric
" Math. Ann. :
Geometry
" Acta Mat.
(1981)
"
texts in Ma-
des courbes
209-230
genera of projective
240, 213-221
" Deformations
de
~ c$ne de Cohen-Macaulay
52, Springer-Verlag
vol. 3__4, 37-42,
E. BALLICO
dans
: " Algebraic
rationnelles
56100
Preprint
2
HARTSHORNE,
Scuola Normale
rank "
codimenslon
thematics,
IT2]
General curves of small genus
" Stable vector bundles
Math. Ann. [ ELLING ]
:
are of maximal
(1979)
of space curves
" Arch. Mat.
(1980)
Ph. ELLIA Superiore PISA
C.N.R.S. Universit~ D~partement
Italy
LA 168 de Nice
de MATHEMATIQUES
Parc
Valrose 06034
NICE CEDEX
VARIETES
RATIONNELLES
ET U N I R A T I O N N E L L E S
A. B E A U V I L L E
Centre
de M a t h ~ m a t i q u e s de l'Ecole P o l y t e c h n i q u e F 91128 Palaiseau C e d e x - F r a n c e
" L a b o r a t o i r e Associ6 au C. N. R. S. No 169"
1.
ENONCE DU PROBLEME.
Le probl~me de L~ro~h. qu'une corps
dont
Ii peut
extension des
pure
rationnelles
sous-extension
Ce p r o b l ~ m e
:
d'une
est en fair
d~finitions
D~finition a)
est souvent
en termes
appel~
alg~briques
si elle
le p r o b l ~ m e
(rappelons
est C - i s o m o r p h e
sur C en un hombre
fini
au
d'ind~-
:
Toute
deux
parler
de C est dite
fractions
termin~es)
je veux
s'exprimer
extension
de nature
de C est-elle
g~om~trique.
pure
?
Introduisons
:
Soit
X une vari6t6
al~6brique
On dit que X est u n i r a t i o n n e l l e
rationnelle
pure
dominante
(c'est-a-dire
s'il
complexe
existe
irr6ductible.
une a p p l i c a t i o n
~n6riquement
sur~ective)
f : ~ n _ _ ~ X. b)
On d i t que X es___~t r a t i o n n e l l e s ' i l
e x i s t e une a p p l i c a t i o n b i r a
tionnelle f : ~n__4 X.
Puisque fonctions
toute
d'une
formulation Toute
Remarques application rale
de p n
dominante. ce que nous
M568.0582
extension
vari6t~
g~om~trique vari~t~
:
l)
suivante
II est
facile
de voir
au moins
la d 6 f i n i t i o n
ferons
d~sormais.
de ~ e s t
probl~me
est-elle
f: F n - - A x
, de dimension
fini le
le
corps
des
de L f l r o t h a d m e t
la
:
unirationnelle
dominante
Dans
de t y p e
alg6brique~
a)~
rationnelle
que
la r e s t r i c t i o n
~ une s o u s - v a r i ~ t ~ 6gale on peut
?
~ celle donc
d'une
lin~aire
g6n~-
de X~ est encore
supposer
n = dim
(X),
17
2) classe
supposerons
2.
est
donc
clair
que
birationnelle
d~sormais
le
probl~me
de
la vari~t~
que celle-ci
est
n e d ~ p e n d q u e de l a consid~r~e
: nous
lisse.
LE PROBLEME DE L~ROTH EN DIMENSION UN ET DEUX.
L~roth toute est
I1
d'~quivalence
a r~solu
courbe
affirmativement
unirationnelle
alg~brique
(alors
La d ~ m o n s t r a t i o n
est
qu'il
donne
g~om~trique
unirationnelle~
de s o r t e
le
est
qu'il
probleme
rationnelle l'~nonc~ tr~s
un
:
Sa d ~ m o n s t r a t i o n
sous
facile
existe
en d i m e n s i o n
ILl.
forme g~om~trique).
: soit
Cune
une application
courbe
f : ~1-*C.
J
L'espace sur
H ° ( C ~ C) e s t
C~ l a
courbe
de g e n r e
Le
l'un
S une
et
:
Th6or~me
de
lisse
Toute
g~om~trie
que
f est
la
surface
lisse
S telle
3.
LE PROBLEME DE LUROTH EN DII~ENSION TROIS.
se
dtailleurs
unirationnelle~ surtout
le
trouvait
d~j~ et
des
observ~
pos~
la
de L ~ r o t h
en dimension
tionalit~
de
la
quadrique
et
d'une
cubique
dans
que cette
derni~re
vari~t~
est
en princ i p e
quartique
de ~ 4
un contre-exemple
et p5
que S est
pos~ en dim e ns ion
3.
:
G. F a n o q u i D~s 1 9 0 8 ,
En 1912~
unirationnelle au p r o b l ~ m e
EEJ,
de L ~ r o t h .
dans
~4
est
Mais cWest reste
Fano
de l ' i n t e r s e c t i o n [F1J.
3 ~ Fax N~ther
cubique
de s a r a t i o n a l i t Y .
italien
f d'une
= 0 pour
de C a s t e l n u o v o
lthypersurface
question
nom du m a t h ~ m a t i c i e n
au p r o b l ~ m e
lors
que
nombre
par
que HO(s,~)I = HO(s,(~)@2)2 = 0
rationnelle.
Le p r o b l ~ m e
applica-
d'un
inverse
entra~ne
essentiel
est
avait
une
a
on a H ° ( S , ~ )® k ) propri6t6
r6sultat
italienne.
en d e h o r s
l'image
deux.
constitue
f : ~--~S
qu'on
d6fini
de d 6 f i n i r
que cette
le
en dim e ns ion
~ ce r ~ s u l t a t
et
Or t o u t e
r~sultat.
birationnelle
haut
g6n6ralement,
~ montrer c'est
[C]
comme p l u s
permet
~ plus
difficile
1894
forme holomorphe donc ~=0.
le
plus
unirationnelle~
0 (noter
ce qui
une
nulle~
~ ~1 ~ d'ou
en
la
~ est
est
beaucoup
On m o n t r e
I1 reste
rationnelle~
isomorphe
succ~s
si
f ~ sur
Castelnuovo
= H°(S,~)=
de p o i n t s ~
k.
: en e f f e t ,
devient
par
forme holomorphe) tout
est
surface
dominante.
H°(S,~) fini
r~solu
des premiers
Soient tion
nul
probl~me
I1 a ~t~
nul
forme holomorphe
attach~
VVprouve" l ' i r r a complete
Enriques fournissant
d~une
d~montre ainsi
Malheureusement
-
18
l'argument base
de F a n o s e h e u r t e
des
syst~mes
ne p e r m e t t e n t
paraissent
lin~aires~
pas
(implicitement)
aujourd'hui
Dans les
qui
ann~es
~ plong~es
vari~t~s
cities
En 1 9 4 7 ,
il
pretend
£ la
n'est
pas
I1 ses
plus
dans
~n
en constituent
de N ~ t h e r
que
les
contemporains
rues
plus
dans
le
tard
: un expos~
livre
de R o t h
[R].
de F a n o n ' e s t
~ l'abri
de
son tour,un s'agit
"rigoureux" obtenu
d e u x ~ d e u x de 4 h y p e r p l a n s
en position
vari~t~
est
unirationnelle,
de t o r s i o n ~
simplement
connexe.
ou,
le
telle
vari~t~
est
certainement
(pour
Roth),
Serre
d~montrait
vari~t~ L'erreur
unirationnelle de R o t h
des singularit~s
I1 soit le
a en f a i r
r~solu~
et
est
provient isol~es
fallu
:
deux
s~rie
birationnel-
~ il
r~pondrait
d~monstration
~t~
de ce q u e
6 plans,
g~n~rale. que
: il
une sextique intersections
Roth
son groupe
d~montre que de P i c a r d
au m~me~ q u ' e l l e
admet
n'est
pas
birationnel,
une
Malheureusement
ann~es
solide
des 6 plans
1970 p o u r
d~monstrations
de L ~ r o t h
en norma lis a nt par
appa-
trouve
en e x h i b a n t ,
au p r o b l ~ m e
simplement le
des
accept~s
sont
de F a n o s e
poursuit
irrationnelle.
en d e h o r s
largement
Des r ~ s e r v e s
un i n v a r i a n t
quelques
toujours
3 paires
de
(les
de c e t t e
~tant
la
qu'aucune
revient
~1 e s t
attendre
ce p a r
diagramme suivant
puis
ce q u i
Puisque
vari~t~s
exemples).
de s n
travaux
I1
doublement
cette
vari~t~s
EG]).
conclut
~ passer
un ~ l ~ m e n t
nous
anticanonique
types
cubique
des
critique.
d'Enriques",
astreinte
les
premiers
autres
exemple
Roth la
faire
qui
pr~c~dentes.
critique
contre-exemple
du " s o l i d e
dans ~n
les
de F a n o a i e n t
par
points-
a u x m~mes c r i t i q u e s .
syst~me
de c e s
les
r~sultats (cf.
g~n~rale~
-malheureusement
que
les
de l ' ~ p o q u e
~ Fano dolt
longuement
que trois
~ certaines
rigoureuse
semble
leur
~ une hypersurface
question
techniques
pas
Fano ~tudie par
sur
F a n o d o n n e e n 1915 u n e a u t r e
n'~chappe
haut
IF4]
~quivalentes
ainsi
injustifiables.
d~montrer
irrationnels
lement
par
plus
les
de p o s i t i o n
mais qui
suivent,
d~licates
rigoureusement
des hypotheses
[F2]~
sont
auxquelles
de r ~ p o n d r e
IId~monstration"
dimension
~ des questions
que
plus
tard
connexe d'Enriques doubles
le
de m a t h ~ m a t i c i e n s ~
qu'une [Se]
!
poss~de [Ty~ .
probleme
de L U r o t h
sch~matis6s
dans
19
auteurs
Clemens-Griffiths
cubique
Iskovskikh-Manin
quartique
hrtin-Mumford
sp6cial
le
et
de c r i t ~ r e s
pas rationnelle.
m6thodes Iet cette
4.
d'affirmer
~ 4 (cf.
de c e t
3.
d'etre
de H 3 ( X , ~ )
dans ~4)
qu'une
r6solu.
d'un
type
sont
unira-
s e p o s e de d i s -
vari~t~
expos6 est
birationnels
enti~rement
la question
dans une certaine
en d i m e n s i o n
en d i m e n s i o n
loin
de b a s d e g r 6 D~s l o r s ,
Le b u t
2 permettent
question
ouvert
permettant
torsion
montre que des vari6t6s
(hypersurfaces
n'est
interm6diaire
automorphismes
§ 9)
cependant
non r a t i o n n e l l e s .
poser
jacobienne
dans F 4
diagramme ci-dessus
extr~mement simple tionnelles
dans ~4
(cf.
Le p r o b l ~ m e de L U r o t h e s t En e f f e t ,
m6thode
exemple
donn6e est
de m o n t r e r
ou
que les
m e s u r e de r 6 p o n d r e
Le p r o b l ~ m e e s t
par
contre
enti~rement
§ 10).
LES CANDIDATS.
I1 s'agit lisses
ici
X parmi
condition espaces
de d 6 c r i r e
lesquelles
n6cessaire, de t e n s e u r s
une classe
figurent
d'apr~s
ce q u i
contravariants
assez
d'6ventuels pr6c~de
g6n6rale
de v a r i 6 t 6 s
contre-exemples. (§ 2 ) , e s t
holomorphes
soient
Une
que tousles nuls,
c'est-~-
dire
H ° ( X , ( ~ ) ®k) = 0
pour t o u t
k~ 1
Cette condition d6crit une classe int6ressante de vari6t6s, mais certainement trop large pour qu'on puisse esp6rer une classification. I1 est raisonnable,
pour simplifier,
d'ajouter
la condition b 2 = 1
(6quivalente ~ P i c ( X ) = ~ ) . On obtient alors les vari6t6s de Fano de premiere esp~ce, qui ont 6t6 classifi~es par Iskovskikh ~I 1,2]. Pour 6noncer ses r6sultats~ a K X = -rH X ~
a alors
o~ r e s t
notons H X le g6n6rateur ample de Pic(X) un entier > 0
; on
qu'on appelle l'indice de X. On
20
:
Th6orbme
(i)
Soit
X une vari6t6
On a r ~ 4 (resp.
; sir
de Fano de p r e m i b r e
= 4 (resp.
~ une quadrique
(ii)
S_~i r = 2~ X e s t
(A 1)
h~persurface
r = 5)~ X e s t
lisse
isomorphe
de d e s r 6
esp~ce,
d'indice
r.
isomor~he a ~3
de ~ 4 ) .
~ l'une
des vari6t6s
6 dans l'espace
pro~ectif
lisses
suivantes
:
quasi-homo~ne
~(1~1~1~2~3) (A 2)
rev~tement
(A 3)
cubique
(A 4)
intersection
(A 5)
section
(iii)
d o u b l e de ~ $
isomorphe
= 1, X e s t
rev~tement
(B 4 )
quartique
double double
le
d'une
quadrique
d'une
(B 8 )
de t r o i s
section
quadrati~ue
(B14)
section
lin~aire
Les r ~ s u l t a t s
Proposition
Toutes
Certaines quant nelles
sur
d'une
suivantes
s exti~ue
quartiques
sont
de d e s r ~
sont
d dans ~d/2+2 ou l ' u n i r a t i o n a l i t ~
essentiellement
de F a n o de l ~ r e
t~pes
affirme
de l a q u a r t i q u e qu'elles
dus ~ Fano :
esp~ce sont unira-
A1, B2 e t B4 .
unirationnelles
Certaines
dans p5
;
la rationalit~
les
cubique
le lon~
de p 4
dans ~6
de t ~ p e A4~ A5 ; B12 ~ B16 ~ B18 s o n t
~ Iskovskikh
[Sg]
g6n6rique,
vari~t6s le sont
rationnelles. ~ on i g n o r e ainsi
tout
que d e s
de t y p e B22 s o n t
toutes~
ration-
mais sa d~monstra-
incomplete.
Ces r ~ s u l t a t s : par
(type
lisses
;
~ ils
les vari~t~s
(peut-~tre)
de t y p e A1 e t B2 .
est
et
de G ( 2 t 5 )
classiques
~ l'unirationalit~
vari6t~s
lon~ d'une
de l a
Q d_~e ~ 4 ~ r a m i f i 6
quadriques
une vari~t6
positifs
sont
quadrique
de G ( 2 ~ 6 )
(d= 12,16~18~22)
de c e s v a r i 6 t 6 s
ques
ramifi~
d6coup6 sur Q par une quartique
(BIo)
tion
de p 3
intersection
Les v a r i ~ t ~ s
de P l f l c k e r )
des vari6t6s
intersection
:
exemple~ A4)
;
dans ~4
diviseur
saul
quartique
dans ~5 ;
~ l'une
(B 6 )
tionnelles
lon~ d'une
le plon~ement
Sir
rev~tement
p5
(dans G(2,5) .
(B 2)
(B d)
le
de deux q u a d r i q u e s
lin6aire
~rassmannienne
d'un
ramifi6
dans ~4 ;
se d~montrent soit
par des m~thodes projectives
X une intersection
~ la projection
depuis
de deux q u a d r i q u e s
une droite
contenue
classidans
dans X
:
21
d~finit loin
5.
une a p p l i c a t i o n
birationnelle
(§ 6) une d ~ m o n s t r a t i o n
de X d a n s ]p3 . Nous v e r r o n s
de l ' u n i r a t i o n a l i t ~
plus
d e s t y p e s A~ e t B8 •
LA JACOBIENNE INTERMEDIAIRE.
L'outil utilis~ par Clemens-Griffiths [C-G] est la jacobienne interm~diaire.
Je me bornerai
~ la d~finir dans un cas ires particulier,
celui d'une vari~t~ X de dimension 5 (lisse~ projective) v~rifiant H°(X~) = O. Dans ce cas la d~composition de Hodge de H 3 ( X , [ ) s ' ~ c r i t
simplement H3(X,[)
o~ H2 ' 1 e t H1~2 s o n t (on c o n s i d ~ r e
deux s o u s - e s p a c e s
H3(X,[)
complexe).
que l a p r o j e c t i o n
de H3(X~Z)
H1~2/H3(X~)
interm~diaire
,
c o m p l e x e s c o n j u g u ~ s de H3(X~C)
comme l e c o m p l e x i f i ~
d'une conjugaison quotient
= H2',I~HI' 2
est
Cette
d a n s H1~2 e s t
un t o r e
de X. De p l u s ~
de H 3 ( X , ~ ) , c e q u i
propri~t~
de c o n j u g a i s o n un r 6 s e a u ~
complexe J(X),
la forme ( a , ~ ) ~ - 2 i
le munit
entra~ne
de s o r t e
que l e
appel~ ~acobienne ~
~A~
s u r H1 ' 2
X possede les c'est
propri~t~s
suivantes
une ~ r m e h e r m i t i e n n e
sa partie
imaginaire
une f o r m e a l t e r n ~ e Elle autrement
d~finit dit
une v a r i ~ t ~
induit
par consequent th~ta,
La j a c o b i e n n e
ab~lienne
s~parante
sur H3(X~)
unimodulaire
un d i v i s e u r
p a r e x e m p l e EM1J).
:
positive
(~ v a l e u r s
s u r H1~2 •
le cup-produit~
c'est-~-dire
entieres).
s u r J ( X ) une p o l a r i s a t i o n
principale~
bien
pros
d~fini
~ translation
interm~diaire
principalement
polaris~e~
est
(cf.
donc d a n s c e c a s
et c'est
toujours
ainsi
que n o u s l a c o n s i d e r e r o n s .
La construction pr~c~dente est strictement parall~ie a celle de la jacobienne d'une courbe ~ de fait la jacobienne interm~diaire joue~ pour les vari~t~s qui nous occupent~
le m~me r$1e fondamental que la
jacobienne pour les courbes. Je me contenterai
de mentionner ici d'une
part qu'elle intervient dans l'~tude des cycles de dimension un sur X~ et d'autre part qu'on peut esp~rer qu'elle d~termine la vari~t~ X (probl~me de Torelli).
Hais son importance dans les questions de ra-
tionalit~ vient du r~sultat suivant (EC-GJI cor. 3.26)
:
22
Proposition une
:
Si
Oacobienne
Par part
il
la
vari~tb
X est
ou u n p r o d u i t
jacobienne s'agit
on e n t e n d
bien
sSr
rationnelle,
J(X)
est
isomorphe
de j a c o b i e n n e s .
d'un
la
jacobienne
d'une
isomorphisme
courbe
de v a r i ~ t ~ s
~ d'autre
ab~liennes
pola-
ris~es.
Es~uisse logique
de d ~ m o n s t r a t i o n facile,
jacobienne et
que
lorsqu'on
interm~diaire
de J ( B ) .
Si
: On v ~ r i f i e
X est
~clate
d'abord~
une
de l a v a r i ~ t ~
rationnelle,
il
par
courbe
~clat~e existe
un c a l c u l
lisse est
B d ~ n s X~ l a
le
d'apr~s
cohomo-
produit
Hironaka
de J ( X ) un diagramme
R
IP 3
ou ~ e s t lisses, J(R)
compos~ d'un et
est
ou ~ e s t
un p r o d u i t
--~ X
nombre fini
d'~clatements
un morphisme birationnel. de j a c o b i e n n e s ,
et
J(X)
de p o i n t s D'apres
en e s t
Or u n e v a r i ~ t ~
ab~lienne
principalement
polaris~e
mani~re
en p roduit
de f a c t e u r s
irr~ductibles~
unique
correspondant proposition (leur
aux composantes r~sulte
diviseur
Le p r o b l e m e le
probl~me
liennes
crit~re le 4 dans
les
Parmi facile
commode
polaris~es,
que les
les
ainsi
d~pendent
le
probl~me
propri~t~s
du t h ~ o r ~ m e
des
se
facteurs th~ta.
sont
th~or~me
un autre
parmi
les
La
irr~ductibles
de R i e m a n n ) .
probleme
routes
direct.
d ~ c o m p o s e d__~e ces
jacobiennes le
pr~c~de~
un f a c t e u r
du d i v i s e u r
par
: d~terminer,
premieres
de s o r t e .
de c e q u e irr~ductible
de L f f r o t h r e j o i n t
principalement
de 3 g - 3 ~
s~quence
est
de S c h o t t k $
Notons que
que g~4
alors
th~ta
irr~ductibles
ou de c o u r b e s
ce q n i
classique~
vari~t~s
ab~-
celles qui sont des jacobiennes. 1 de ~ g ( g + l ) - m o d u l e s et les secondes est
(hautement
!)
non trivial
g~om~triques
des
jacobiennes,
singularit~s
de R i e m a n n f o u r n i t
des une conun
:
lieu
sinsulier
la
jacobienne.
du d i v i s e u r
~
d'une
~acobienne
est
de c o d i m e n s i o n
23
hndreotti loin
Mayer
ont
prouv6
de c a r a c t $ r i s e r
les
On n o t e r a
produit
r6ductible, vet
que
que
le
poser
qu'un
qui
la
a donc
varibt6
diviseur d'une
r6pondre t6s
et
~
jacobiennes,
de F a n o .
lesquelles
cf.
de v a r i 6 t ~ s
ab~liennes
X n'est
pas
rationnelle,
de J ( X )
est
peu
singulier.
dbom6trique
de J ( X ) .
question
telle
singulier
en g$n6ral, d~crire
propriSt$
pas
tr~s
a un diviseur
de c o d i m e n s i o n
2.
on c h e r c h e r a I1
faut
On e s t
ne serait-ce
maintenant
description
n'est
[A-M].
lieu
Nous a l l o n s
une
cette
un
description
~ cette
que
une
Pour
donc
pour loin
que
cela
dis-
de s a v o i r
pour
classe
prou-
a montrer
les
vari6-
de v a r i 6 t 6 s
pour
existe.
t
6.
FI BRES EN CONIQUES.
D~finition
:
existe
une
fibres
sont
I1 tion
On d i t
surface des
est
coni~ues
facile
rationnelle
sur
(ou,
let
de c o u r b e s
un qui
fibr~
Sous qu'il -
-
si
en
les
-
sip
est
pest
On d i t
cependant
le
Proposition surface
est
Soit
rationnelle
la
tout
r~sultat
la
une
s'il
dont
une
les
une
une
applica-
courbe
rationnelle
birationnellement
radu
~quivalente
classe
de vari~t~s
rationnelles.
on v ~ r i f i e
facilement
que
courbe
rationnelle
f-l(p)
est
lisse
r~union
de
2 courbes
est
droite
transversalement de C,
courbe
que
est
congruence
doric a i n s i
de C,
double
f : X~S
en c o n i q u e s f : X~S
admettant
g~n~rique une
d~finition,
f-l(p)
ordinaire
coniques
est
d~ £ E n r i q u e s
un la
fibr~
une
double
;
de C •
discriminante
en
suivant~
telle
vari~t~
fibre
telle
fibr~
flbr~e
d~n~r~es).
est
singulier
un point
si
:
de CoS
coupant
que C est
On i g n o r e
la
d'etre
lisse
un p o i n t
alors
route
On d ~ c r i t
f-l(p)
se
X est
un m o r p h i s m e
classique,
proches"
un p o i n t
rationnelles
que
rationnelles)
courbe
C,
est
Set
dont
hypotheses une
pE S-
sip
F 2
langage
"assez
existe
vari~t~
(~ventuellement
en coniques.
semblent
la
de v o i r
tionnelle ordre
que
rationnelle
du
en coniques. (cf.§
10).
:
en coniques,
restriction
fibr~
unirationnel
et
soit
de f ~ R s o i t
R~X
une
surjectiv~,
On a
24
de de~r~ d.
Alors la vari~t~ X est unirationnelle ; plus pr~cis~ment~
i_~l existe une application rationnelle dominante ~ 3
D~monstration
:
rique
une
de f e s t
rationnel
sur
Traitons
K ~ elle
st~r~ographique g~n~ral
s'en
Exemples
1)
une
tion
de
La c u b i q u e
diviseur
f a E est
2) Soit de ~ 6
de ~ 6 couples
(q,~)
tels
seulement
si
d~terminant est
de d e g r ~
N o t o n s X£ l a
soit
X£ l a
l'bclatement 2.
Ainsi
projectif.
dans
de c e n t r e
courbe d~finie
7~ £ c o e f f i c i e n t s
une
~ ~4)
f : Q£(X)~g
dans une quadrique
restric-
de q u a d r i q u e s
des
de ~ x G ~
que
la
~6.
r~seau
(isomorphe
la
;
unirationnelle.
Choisissons
sous-vari~t~
est
obtenue
en c onique s .
de £
X est
Notons W le
que
vari~t~
5 .
imm~diatement
de s o r t e
la
droite 2-plans
form~e fait
des
de Q ~ ( X )
sous-vari~t~
gne par
<~x>
plan
singuli~re
si
C est
l'ensemble
des
par
l'annulation
lin~aires,
d'un
et
par
obtenue
en ~ c l a t a n t
~ d a n s X~ e t
des couples
(q~x)
que <~x>cq
engendr~
par £ et
tels
x ; ce s ym bole
S la
a un s e n s
On a u n d i a g r a m m e
avec a(q,x)= (q~<~,x>)
Q~(x)
II est imm~diat que P2 e t a
sous-
(on d~si-
S
X£
de
q est
7. vari~t~
form~e
le
Le c a s
La p r o j e c t i o n
La p r o j e c t i o n
d'ordre
de ~ × X £
x~X~).
.
la
de ff ~ e l l e
vari~t~
tout
Q£(X)
contenus
sym~trique
projection
de X£ un f i b r b
de d e g r ~
G£ l a v a r i ~ t ~
q l'est~
g~n~-
R~S.
de 3 q u a d r i q u e s
On v ~ r i f i e
singuli~res
fait
dans
un plan
que ~Cq
fibre
admet un p o i n t
rationnelle.
de b a s e
de d e g r b
de c e t y p e .
Soit
des plans
quadriques
est
exceptionnel
en c o n i q u e s .
G£ f o r m ~ e
qui
CCp2
surjective~
~ .
(par
dans X .
X£ ~ 2
dans X ; notons
un f i b r ~
suite
~ contenue
X ; c'est
contenant
~ ~K1
dans p4 ~ et
L'intersection
contenant
la
qui
que X est
de desr~ d.
de p 4 .
(lisse)
X une vari~t~
contenue
et
entra~ne
le
f:
d = 1 . Alors
K= E ( x , y ) ,
donc K - i s o m o r p h e
changement
droite
cas
corps
par
diseriminante
Notons E le
le
le
ce q u i
un morphisme
La c o u r b e
sur
est
X une cubique
en b c l a t a n t d4finit
!), d~duit
:
Soit
d'abord
conique
_)X
sont des morphismes birationnels~
de
pour
25
sorte
que X est
birationnellement
6quivalent
au f i b r 6
en c o n i q u e s
Qz(x). De n o u v e a u projette
le
diviseur
surjectivement
exceptionnel
sur
ff ( l e
dans
degr~
est
l'6clatement
4).
Ainsi
de £ s e
X est
uniratlon-
nelle.
3)
Autres
Les vari6t6s coniques bles
; mais
si
exemple -
elles
on l e u r (cf.
le
le
permet
[B1])
celle-ci
l~re
esp~ce
deviennent
d'acqu~rir
par des
ne s o n t des
-
la
l'intersection
des
fibr6s
sp6cialisations
singularit~s.
double
de $ 3
ramifi6
le
a d m e t au m o i n s u n p o i n t
-
pas
en
convena-
Citons
par
:
revStement
lorsque
exemples.
de F a n o de
quartique
avec
une
droite
darts ~ 5
double
d'une
long
double
d'une
quartique~
ordinaire
;
;
quadrique
et
d'une
cubique
contenant
un p l a n .
L'importance nalit$
vient
Th$or~me courbe
:
Soit
fibr6s
C est
( o u un p r o d u i t pas
d'autre
([M2]~
de c e
s'assurer
montr6
tionnelles.
pour
qu'une
exceptions~
Corollaire
f-l(p)
sur
n'est
les
questions
~2.
pas
Si
le
de r a t i o -
de~r6
isomorphe
En p a r t i c u l i e r t
que
:
le
courbe ce q u i
la
consiste courbe
correspondant pour
pE C .
de l a
~ une
~acq-
la variSt6
X
le
cas
lieu
Les vari6t6s
pour
de d e g r $
(cf.
g6o-
®
des
que J(X)
[BI~)
du d i v i s e u r un pe tit
o5 C a d e s p o i n t s
plane est
~ d~crire
aux deux composantes
£ (~,C)
singulier
~ 5 sauf
d'abord
C est munienaturellement
Mumford a p r o u v ~
de P r s m a s s o c i 6 e
de c o d i m e n s i o n
[B2]
que
double C~C,
£ la vari6t6
part
et
J(X)
th6or~me
On o b s e r v e
singuli~res
de Prym e a t
ces
J(X).
rev~tement
isomorphe
en coniques ~ 6,
de 3 a c o b i e n n e s ) .
La d 6 m o n s t r a t i o n
coniques
pour
:
rationnelle.
m6triquement d'un
en coniques
suivant
X un fibr6
discriminante
bienne n'est
des
du t h 6 o r e m e
; il
d'une
est
a vari6t6
nombre d'exceptions
doubles).I1
reste
au m o i n s 6 ne r e n t r e
pas
dans
facile.
de F a n o de t y p e
A3,
B8 e_~t B14 s o n t
irra-
26
Pour exemples
les
types
h3 et
1 et
2 (3e
triche
discriminante th6or~me
est
s'6tend
B8 ,
de d e g r 6 ~ ce
5 ; une
cas).
birationnellement
7.
IRRATIONALITE GENERIQUE.
La m 6 t h o d e
Th6or~me [BI]
:
B~,
De m a n i ~ r e de c h a q u e
que
type
vari6t6
biennes
existe
suivant
il
:
montrer
Soient
clair
et
des
la
courbe
montre
vari6t6 de ~ 4
que
le
de t y p e
B14
IF3].
permet
de r 6 g l e r
:
8tre
que
de t y p e
par
en f a i r
hl,
h2,
B2,
B4,
que
la
distincte
irr6-
de T,
telle
irrationnelles.
jacobienne
jacobienne
est
des vari6t~s
par une vari6t6
Z de T,
T- Z soient
pas une
propri6t6
l'ensemble
param6tr6
ouverte~
La d 6 m o n s t r a t i o n
-
X n'a que o la vari6t6
s]
n'est il
S une courbe
de d i m e n s i o n
pour
o~
pas
la
fibre
des
interm6diaire
ou u n p r o d u i t il
repose
lisse
:
le
suffit
alors
de j a c o -
de l e
sur
le
prouver
lemme
doubles
£ une
et
famille
de S .
de v a r i 6 t 6 s
On s u p p o s e
H°(Xs~Q;
que
:
)= O ;
s ; s i X' d S s i ~ n e -o ces points doubles~ J(X') o ou u n p r o d u i t de O a c o b i e n n e s .
ordinaires
en 6clatant Oacobienne
non vide
une
U de S tel
de ~ a c o b i e n n e s )
que J(X u)
ne s o i t
pour
uE U .
tout
pas
Notons a
groupe
J(Xs)sES_ ° d6finit sur
lisse~
(ou un produit
polaris~es
par
o un point
un ouvert
principalement
X~S
trois~
obtenue
D~monstration
lisse,
Xs e s t
points
isomorphe
existe
Oacobienne
th~se
interm$diaires
est
peut
de T - Z n ' e s t
-
Siegel
pr6cise
qu'une
de F a n o ~ 6 n 6 r i q u e
param6tr6es
de T - Z.
projectives
une
dont
:
Lemme
hlors
plus
£ une cubique
une sous-vari6t6
; comme c e t t e
en u n p o i n t
6rude
du t h 6 o r ~ m e cubique,
irrationnelle.
consid6r6
vari~t~s
la
en suspens
Une v a r i 6 t 6
Nous allons d'une
encore
pr6cise,
T ; il
les
jacobiennes
cas
B 6 o_~u BIO e s t
ductible
6quivalente
des
les
r6sulte
Fano a montr6
est
$6n6riquement
cela
un p e u p o u r
X entra~ne
l'espace des modules des vari6t6s ab61iennes g de d i m e n s i o n g ( q u o t i e n t du d e m i - e s p a c e de 1 modulaire), a v e c g = ~ b 3 ( X s ) . La f a m i l l e
une
application
que c se
classifiante
prolonge
en ~ :
c : S- O~g S~g,
;
l'hypo-
ou ~ g d ~ s i g n e
la
27
compactification
de S a t a k e
= ~
g et
le
point
p= dim J(X') o correspondant est
fermb
~(o)
de ~
~g.
Soit
aux
dans
a
g
de a g -
est
g
Ua
g la
son
adherence
= J
g
U .-.
UJ
g
Ua
,
o
de J ( X ' ) dans ~ ca , avec o p g part J l'ensemble des points de ~g g et aux produits de j a c o b i e n n e s ; il
d'autre
et
g-1
classe
jacobiennes ,
On a e n s e m b l i s t e m e n t
dans a
g-1U
"'"
UJ
g
est
o
Comme ~ ( 0 ) ~ par hypoth~se, il existe un voisinage g de a dans S tel que ~(U) N ~ = D ~ d'ou le lemme. g
Pour
achever
en 6vidence~ propri6t~s
intersection la
lemme,
de
que
projetant type point
double
B6 ~
cas~ telle
peut
depuis
et
une
X~S
droite
en une
un point
acqu~rir
ordinaire
d~formation
une quartiqueX
d~former
depuis
faisant
une
il
reste
~ mettre
poss~dant
les
X' s o i t u n f i b r ~ e n c o n i q u e s . Je o quelques exemples. Si l~on projette une
on o b t i e n t l'on
du t h ~ o r ~ m e ~
que
3 quadriques
de n o u v e a u
B2 ; e n
et
d'indiquer
vari6t6,
ordinaires
d6monstration
darts chaque du
me c o n t e n t e r a i
dans
la
ouvert U
g~n6rale
° avec
contenue
17 p o i n t s
quartique
doubles
g~n6rique.
En
double
de X , on o b t i e n t o ~ l'intersection de 3 q u a d r i q u e s
en projetant
depuis
ce
point~
le
le un
type
etc.
Remarque
:
La m 6 t h o d e
que
les
vari6t6s
8.
LES TRAVAUX
Les
Th6or~me
D'ISKOVSKIKH
r6sultats
:
s'applique
bien
entendu
a d'autres
vari6t~s
de F a n o .
d'Iskovskikh
Les vari6t~s
ET MANIN.
et
de F a n o
Martin s o n t
de t y p e
B2,
les
suivants
B 4,
B~,
:
B6 s o n t
irra-
tionnelles.
Les celles
de
d6monstrations [F2])
en
les
reprennent compl6tant.
les Elles
id6es sont
de F a n o longues
(essentiellement et
difficiles~
28
et
j'avoue
ne p a s
~nonc~s qui
Pour use vent
dans
de l e r e
(cf.
aussi
application
espece
est
fini~
et
car
la vari~t~
sont
pas
le
~ indiquer
Iskovskikh
les
et
Manin p r o u -
: de X d a n s u n e v a r i ~ t ~
automorphismes
compliqu~e
alors
des
de F a n o
types
esp~ce,
l'~nonc~
existe
B6 ,
birationnels dans
qui
[I3]
ne
consiste
~ notant
suivant
birationnelle
il
de X
~ ~3 . B~ e t
automorphismes
obtient
une application
de F a n o de l e r e
les
d'Iskovskikh
de c e s il
birationnels
~quivalente
pour
automorphismes
La m ~ t h o d e
certains
engendrent~ est
des
birationnellement
plus
d~finis.
qu'ils
[I5])
groupe
pas
X admet
S~i X : X ~ V
-
vari~t~
le
en ~v i d e n c e
groupe
me b o r n e r
B 2 ou B4 ~
birationnelle
est
partout
mettre
Je vais
un isomorphisme.
X n'est
La s i t u a t i o n
lu.
X de t y p e
En p a r t i c u l i e r ~ est
tout
au t h ~ o r ~ m e .
vari~t~
[I-M]
Toute
-
avoir
conduisent
B(X)
:
de X d a n s u n e
, E B(X)
tel
~ue X ° * s o i t
~n i s o m o r p h i s m e . I1 en d ~d u i t ainsi
que
automorphismes
aux fibres
Th~or~me IS] criminante nel
pour
:
terminer
Soit
C v~rifie
X~S
un fibr~
fibres alors tels
9.
en c o n i q u e s une
courbe
exemples
e_~t X e s t
est dans
X irrationnel elliptique).
avec
Y = ~3)
du g r o u p e
s'appliquent
en c o n i q u e s , Alors
fibration,
des points
de X ( p r e n a n t structure
des
avec
:
P o u r S = ~ 2 , ce r ~ s u l t a t en ~ c l a t a n t
la
que ces m~thodes
14K S + C[ ~ .
la
sur
de X .
en coniques
de X p r e s e r v e
mais
l'irrationalit~
profonds
birationnels
Signalons succ~s
en particulier
des renseignements
tout
moins fort
I1 n'est
la.courbe
automorphisme
dis-
biration-
irrationnel.
que
p2 ~ Sarkisov avec
dont
le
H~(X,Z) = 0 pas
th~or~me
donne
clair
du § 6
des exemples
(la
de
courbe
C est
qu'il
existe
de
de c e t
expose,
X unirationnel.
L'EXEMPLE D'ARTIN-MUMFORD.
La m ~ t h o d e ne s ' a p p l i q u e
de J A r - M ]
pas
sort
aux vari~t~s
u n p e u du c a d r e de F a n o
~ elle
a cependant
puisqu'elle
l~avantage
29
de f o u r n i r bas~e
un crit~re
sur
le
r~sultat
Proposition
:
de t o r s i o n
d'irrationalit~ suivant
Pour
est
La d ~ m o n s t r a t i o n mais
plus
I1 pour une
s'agit
si
une
G la
droite
congruence
de R d a n s
G .
f-l(q)
cation
f se
des
droites
de R e ~ e
dans
X.
proposition
du § 5 ,
X la
de t o r s i o n .
X Voici
Soit projective
suivantes
S,
:
pas
droites
contenues une
(cf.
une quadrique
de ~ 3 ; s o i t dans
surface
[B3~,
p.
~ l'ensemble
de ~ ,
£ .
R la
un pinceau
d'Enri~ues, 156).
sous-
de q u a d r i -
appel~e
classi-
Posons
I ,¢cq}
A projection G~G n'est autre ^ f : G~g la seconde projection
factorise
des
que
l'~clatement
; pour
g~n~ratrices
qEg
de q .
,
la
L'appli-
donc en
: ~
double,
aux quadriques Sl,...,Slo
g ~X'
(lisse)
facilement
~%TT
ramifi~
le
singuli~res.
long
est
obtenue que X est
de l a
Celle-ci
,correspondant
i ) = { p i ~, o5 P i
vari~t~
pour
{(£,q)EGxIT
s'identifie
On v o l t
unirationnelles
la
Notons
ordinaires
des
que Rest
que
de ff ~ o n a ~ - l ( s On n o t e
sous-groupe
d'Artin-Mumford.
de U n e c o n t i e n n e n t
un rev~tement
correspondant doubles
le
g~n~rale
singuliere
f
ou ~ e s t
le
de p 3 , de d i m e n s i o n
grassmannienne
On s a l t
imm~diat
fibre
de
~l~ments
l'exemple
de p o s i t i o n
:
I1 est
X,
de p o i n t - b a s e
de G f o r m ~ e
de U .
de
des
de q u a d r i q u e s
quadriques
Notons
quement
lisse
des vari~t~s
contienne
conditions pas
~ est
autres
ques
est
birationnel.
~ celle
de c o n s t r u i r e
g~om~trique
les W n'a
vari~t~
analogue
H3(X,~)
lin~aire
v~rifiant
les
donc
description
un syst~me
(ii)
Elle
facile.
lesquelles
(i)
projective
un invariant
est
dimension.
:
une vari~t~
de H3(X~2E)
en route
un point
a
quartique dix
points
aux quadriques double
en ~clatant unirationnelle
les
de ff
de r a n g
ordinaire points (cf.
2
de X ' . pl,...,pl
§ 6,
0
exemple
3).
30
Proposition
:
Le g r o u p e
D~monstration dessus
:
de Pi
H3(X,TZ.)
contient
un ~l~ment
Nous noterons
Q. l a d i v i s e u r 1 U = X' - { p l , . . . ~ p l o ~ et
; on p o s e
g : V~U est -1 Pi = g (pi)
une
fibration
est
ia
cohomologie
eonsid~r~e
en droites
r~union
exceptionnel
ce qui
suit
Le m o r p h i s m e
tandis
se
coupant
est
toujours
2.
de X a u -
V= g-l(U).
projectives,
de d e u x p l a n s
dans
d'ordre
que
en un point.
La
~ coefficients
entiers. a)
Le g r o u p e
isomorphe
H4(G)
contient
~ H4(G)~H2(R),
et
un ~l~ment cl(N)
est
d'ordre
2 : il
un 61~ment
est
d'ordre
en effet
2 dans
e2(~). b)
I1
enest
i et
des
de m~me de H 4 ( V ) , c H3(P.) 1
relations
~ H4(V) c
H3(p.)
~ cause )H4(G)
= O, H 4 ( P . )
de
la
suite
exacte
~ • H4(p.) 1 i
=~®~o
H2(X ) e s t s a n s t o r s i o n c) Le g r o u p e H 2 ( U ) I e s t s a n s t o1r s i o n ; en effet e puisque X est simplement connexe~ et H2(U) est un sous-groupe de H2(X) c puisque HI(Q.) = O. 1 d) Le g r o u p e H 4 ( U ) c o n t i e n t un ~l~ment d'ordre 2. En e f f e t ~ la suite e e x a c t e de G y s i n p o u r la fibration en spheres g : V~U s'~crit HI(u) C
--~H4(U) C
o~ e d ~ s i g n e
le
celle-ci
annul~e
est
l'~l~ment
d'ordre
2 de H 4 ( U ) . c e) Puisque de H4(X)
cup-produit par
H 3 ( Q i ) = O,
de m~me de H3(X)
avec 2.
2 de H4(V) c
~ celui-ci
le
contient par
g ~ H4(V) C
dualit~
Si
la
) H2(V) C
classe
Im(e)]
provient
,
d'Euler
du f i b r ~
O~ l ' a s s e r t i o n (d'apr~s
groupe
H4(U) est c done un ~l~ment de P o i n c a r ~
et
est
c))
d'un
isomorphe d'ordre par
claire
~ sinon
~l~ment
d'ordre
a un sous-groupe 2,
la
en spheres
et
formule
il
enest
des
coefficients
universels.
Remarques et
:
1)
Pour
non rationnelle 2)
dessus fibration
(puisque La v a r i ~ t ~
de U, c e q u i projective
tout
signifie associ~e
n~ l a
vari~t~
H3(X× ~n ~) Vest que
X×
contient
une vari~t~ la
fibration
~ un fibr~
]pn
est
unirationnelle
un ~l~ment
d'ordre
de S e v e r i - B r a u e r g: V-*U n'est
vectoriel
: en effet
au-
pas
la
dans
2).
3~
le
cas
donc
contraire,
irrationnelle.
vari~t~
10.
Un a r g u m e n t
de S e v e r i - B r a u e r
Completer I1 est
style
2)
les
de [ C - G ] ,
Toute
dimension
la
sion
fournisse
question
qu'une
en u n e
souvent
crit~res
le
une
donc ~ t r e
n~gative
est
Brauer
("forme
du t y p e
V/G, sur
> 3,
toutes
rationnelles
h o m o g e n e G/H e s t - i l
de m~me l o r s q u ' o n
a une vari~t~
donn~e.
exp~rimentales
applicables
de d i m e n s i o n
?
aux vari~t~s
r~ponse
la
qui
sont
2 devrait
: vari~t~
X est-elle
rationnelle
Notons qu'on
a construit
probl~mes
semi-simple
complexe V .
?
une vari~t~
?
de S e v e r i -
rationnelle.
les
Si H est
rationnel
de p 5
~ la question
X admettant
un groupe
vectoriel
irrationnelle.
4.
de Z a r i s k i
dans
4 est
u n de c u b i q u e s
, la
de X × ~ I )
souvent
?
de F a n o " s ' a p p r o c h e
en e s t
assertions
~n~ri~ue
irrationnelle
l'espace
de d i m e n -
:
rationnelle,
ou G e s t
rement
I1
doubles
de c o d i m e n s i o n
tordue"
On r e n c o n t r e
de
de
rationnelle
des vari~t~s
~videmment unirationnelle.
au § 9 u n e v a r i ~ t ~
le
int~ressant
irrationnelle
vari~t~
cro~t.
£ ces
en dimension
est
s~rie
en dimension
probl~me
X× ~1
Elle
la
; conjecturalement
Signalonsun Si
d'une
degr~
precis
cubi~ue
famille
rationnelles
plus
dans
?
de p o i n t s
En p a r t i c u l i e r ~u'une
serait
cas,
: d~une vari~ t~
que
quand
Donner des
Prouver
I1
de F a n o . de c h a q u e
IT].
un s e n s
I1 existe
7)
~ X×
que V s'~tend
variSt~s
r~ponse.
(lisse)
donner
usuelles.
6)
montre
d~taill~e
sp~cialisation
rationalitY"
Peut-on
les
irrationnelle
rationnelle
Fano observe la
la
i m p o s e un n o m b r e c r o i s s a n t
5)
~quivalente
de X .
~tude
suivante
d~formation
~ est
pour
qu'une
3 est-elle
Notons
4)
formel
au-dessus
r~sultats
possible
comprendre
de
birationnellement
PROBL~MES OUVERTS.
])
3)
V serait
de m o d u l e s complexe
operant
Ces vari~t~s
un s o u s - g r o u p e
des vari~t~s lin~ai-
sont-elles
f e r m ~ de G,
l'espace
32
L'espace pour
g~ 10.
des
modules
Est-il
Lesprobl~mes encore Yoici
plus
des
rationnel
d'unirationalit6
inaccessibles,
trois
courbes
questions
de genre
? J'ignore
la
semblent
vu
l'absence
classiques,
qui
g est
r~ponse
a l'heure
totale sont
unirationnel des
que g~3.
actuelle
de m 6 t h o d e s
d'ailleurs
existantes.
li~es
entre.
elles.
8)
Donner
un exemple
H O ( x , ( D ~=) ®Ak )
9)
Unequartique
10)
Donner
pour
tout
un exemple
de ~ 4
~4
est-elle
de f i h r ~
possible,
d dans
X non unirationnelle,
telle
Rue
k.
g6n6rique
Un c a n d i d a t de d e g r 6
de v a r i 6 t 6
0 pour
en coniques
sugg6r6
contenant
unirationnelle
une
?
non unirationnel.
par
Enriques,
droite
avec
est
l'hypersurface
multiplicit~
(d-2),
d ~ 5.
BIBLIOGRAPHIE
["A-M]
A.
ANDREOTTI e t
integrals (1967),
on
Soc. [B1]
D. MUMFORD
varieties
[C-G]
:
Ann.
relations Sc.
Norm.
Some e l e m e n t a r y not
for Sup.
abelian Pisa
rational,
examples Proc.
of unira-
London Math.
de P r y m e t
41
(1977),
Surfaces
:
jacobiennes
interm~diaires,
and the
Schottky
problem,
149-196.
alg~briques
complexes,
Ast~risque
(1978).
Annalen
:
Sulla
razionalita
delle
involuzioni
plane,
44 ( 1 8 9 4 ) .
H. CLEMENS e t
P.
GRIFFITHS
:
The intermediate
Jacobian
the cubic t h r e e f o l d , Ann. of Math. 95 (1972), 281-356.
[E]
21
309-391.
Prym varieties
Math.
CASTELNUOVO
Math.
: are
Vari~t~s
10 ( 1 9 7 7 ) ,
A. BEAUVILLE o n 54, S.M.F. G.
On p e r i o d
75-95. :
A. BEAUVILLE Inventiones
[B3]
which
25 ( 1 9 7 2 ) ,
A. BEAUVILLE Ann. E~N.S.
[B2]
:
curves,
189-238.
EAr-M] M. ARTIN e t tional
A. MAYER
algebraic
F. ENRIQUES
:
Sopra una involuzione non r a z i o n a l e d e l l o
s p a z i o , Rend. Acc. Lincei, s . 5 a, 51 (1912), 81-83.
of
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IF1]
G. FANO aventi
:
Sopra
tutti
alcune
i generi
variet~
nulli~
algebriche
Atti
Acc.
a tre
Torino
dimensioni
43 ( 1 9 0 8 ) ,
973-977. IF2]
G. FANO aventi
:
Osservazioni
tutti
sopra
i generi
nulli,
alcune
Atti
varieta
Acc.
non razionali
Torino
50 (1915),
1067-1072. IF3]
:
Sulle
sezioni
rette
dello
spazio
G. FANO
delle Lincei IF4]
11
G. FANO
(1930), :
[G]
[I2]
V.
[I3]
V.
EI-M]
V.
:
Acc.
algebriche
Comm. P o n t .
a tre Ac.
Sci.
non r6solues
de g ~ o m ~ t r i e
alg~brique,
:
Fano
threefolds
I,
:
Fano
threefolds
II,
:
Birational
Math.
USSR I z v e s t i a
11
485-527. Math.
USSR I z v e s t i a
12
469-506.
ISKOVSKIKH
algebraic
varieties,
ISKOVSKIKH e t
counterexamples (1971),
Grassmanniana
Rend.
(19S3).
ISKOVSKIKH
(1978),
variet~
canoniche,
Questions
Paris
ISKOVSKIKH
(1977),
sulle
a curve-sezioni
L. GODEAUX
V.
variet~
635-720.
Hermann, [I1]
della
dimensioni,
329-356.
Nuove ricerche
dimensioni (1947),
spaziali a cinque
J.
J.
automorphisms
Soviet
MANIN
to
the
Math.
:
of
three-dimensional
13 ( 1 9 8 0 ) ,
815-868.
Three-dimensional
Lflroth
problem,
Math.
quartics
and
USSR S b o r n i k
141-166.
vt
ELI
J.
LUROTH
Annalen
:
Beweis
9 (1876),
eines
Satzes
[M1]
D.
MUMFORD
:
Abelian
EM2]
D. MUMFORD
:
Prym varieties
[R]
Academic
Press~
L. ROTH
:
Algebraic
SARKISOV
Mat.
USSR I z v e s t i a
ESe]
J.P.
SERRE
ET]
:
London Math.
B.
SEGRE
:
TYRELL
Soc.
Ergebnisse
34 ( 1 9 5 9 ) ,
Variazione Ann.
Mat. :
manifolds, :
57 ( 1 9 6 1 ) ,
Math.
(1970).
analysis,
Appl.
group
The Enriques
Math.
6,
(1955).
of
of
conic
bundles,
a unirational
ed omotopia 50 (1960),
On d e f o r m a t i o n s Math.
der
York
variety~
481-484.
continua
Pura
897-898.
Press
to
177-202.
fundamental
Soc.
University
Contributions
automorphisms
17 ( 1 9 8 1 ) ,
K. TIMMERSCHEIDT
J.
Oxford
I.
threefolds,
On t h e
J.
rational [Ty]
:
Curven,
(1974).
Birational
V.
algebrica,
rationale
Berlin-Heidelberg-New
IS]
[Sg]
varieties,
New Y o r k
Springer-Verlag,
~ber
163-165.
Annalen
of 258
threefold,
in
geometria
149-186.
threedimensional
(1982), Proc.
267-275. Cambridge
Phil.
CONIC BUNDLES
ON N O N - R A T I O N A L
SURFACES
by M. B e l t r a m e t t i
and P. F r a n c i a (*)
Contents
Introduction §i.
Notations,
definitions,
and p r e l i m i n a r y
§2.
The C h o w group A2(X)
§3.
The a l g e b r a i c
representative
§4.
The c l a s s i c a l
intermediate
§5.
Some o p e n q u e s t i o n s .
results.
of a conic bundle. of A2(X) .
Jacobian
of a c o n i c b u n d l e .
References.
INTRODUCTION
In this paper we s t a t e some bundles study
X on n o n s i n g u l a r
describe,
with negative
the C o n f e r e n c e
A reason
for this of
We m a i n l y of cycles
equivalent
of
to zero
J(X) .
v e r s i o n of a talk
on June
conic
in the c l a s s i f i c a t i o n dimension.
"Open P r o b l e m s
(Italy),
concerning
the g r o u p A2(X)
Jacobian
is an e x p a n d e d
Ravello
S.
2 w h i c h are a l g e b r a i c a l l y
and the i n t e r m e d i a t e
This
occur
Kodaira
up to i s o g e n i e s ,
codimension
(*)
surfaces
is that conic b u n d l e s
threefolds
results
2,
presented
in A l g e b r a i c
1982.
at
Geometry",
35
In the first results where
s e c t i o n we r e c a l l
on conic bundles:
the case S = ~ 2
some r e l a t i o n s
the m a i n
is studied.
reference Moreover
between conic bundles
negative Kodaira In S e c t i o n
for this
is
[B]
we p o i n t out
and t h r e e f o l d s
with
dimension. 2 we find a d e c o m p o s i t i o n
up to i s o g e n y , as d i r e c t the P r y m v a r i e t y
of the g r o u p A2(X),
sum A 2 (S) @ A 1 (S) S PX' w h e r e
associated
s u r f a c e S w i t h q(S)
w i t h X.
= pg(S)
= O
one has the i s o m o r p h i s m A2(X) well
some w e l l k n o w n g e n e r a l
In p a r t i c u l a r
for a
(and not of g e n e r a l
= PX and the t h e o r y
PX is
type)
runs as
as in [B]. In S e c t i o n
is c o n s i d e r e d . induces
prove
We prove
J(X) that
is c a n o n i c a l l y there exists
one sees polarized
to the i n t e r m e d i a t e isogenous
an isogeny
However,
that ~ i n d u c e s
under
of A2(X)
morphisms.
to A x.
Jacobian
~ Pic°(S)
of p r i n c i p a l l y
the a s s u m p t i o n
an i s o m o r p h i s m
J(X)
J(X) .
M o r e o v e r we
c:J(X)~Alb(S)
case a is not a m o r p h i s m
abelian varieties.
A x of A2(X)
that the d e c o m p o s i t i o n
4 is d e v o t e d
In the g e n e r a l
A
representative
the one of A x, via c a n o n i c a l
Section First,
3 the a l g e b r a i c
q(S)
• Px"
polarized = O,
= PX of p r i n c i p a l l y
abelian varieties.
list of some o p e n q u e s t i o n s
Here we give this m a y r e s u l t
a rather
in r a t h e r
is c o n t a i n e d
elementary
exposition
a l e n g t h y paper.
in S e c t i o n
5.
even though
36 § I.
Notations,
definitionsq
Throughout
this
paper we
field k of c h a r a c t e r i s t i c irreducible
nonsingular
For a c o h e r e n t denote
by hi(F)
i a O.
We call
Moreover
scheme
defined
irregularity
over k.
of the k - v e c t o r of V the i n t e g e r
of V is Pm(V)
variety
V we
space Hi(v,F), q(V)
= h I ( O v) .
= h O ( ~ v~m),
where
~V
sheaf.
We d e n o t e by ~(V) the K o d a i r a d of V and by X(O v) = ~ (-l)ihi(c9 v) the Euleri=O
characteristic
of V.
let sr(~PV) be the r - s y m m e t r i c p-forms
closed
we m e a n a c o m p l e t e
F on a d - d i m e n s i o n a l
the m - p l u r i ~ e n u s
dimension
Poincare
By v a r i e t y
sheaf
results
fix an a l g e b r a i c a l l y
zero.
the d i m e n s i o n
is the c a n o n i c a l
and p r e l i m i n a r y
Finally, tensor
for all p, O < p < d,
of the sheaf of r e g u l a r
~P.
By t h r e e f o l d projective
variety
DEFINITION
i.i
(resp.
of d i m e n s i o n
Conic
This
is clear
from
the
three
following
statements
f:X ÷ S such
that
D
on a n o n s i n g u l a r
from the r e c e n t
two).
X is a conic b u n d l e
S and a m o r p h i s m
of t h r e e f o l d s
a nonsingular
(resp.
are conics.
bundles
classification
we m e a n
We say that a t h r e e f o l d
if there exist a s u r f a c e all the fibres
surface)
with
surface
negative
results
occur Kodaira
of Mori
contained
in the
[Mo],
in [B-F].
dimension. and also
37
THEOREM
1.2
L e t X be a t h r e e f o l d
with
K(X)
< O,
q(X)
> O.
T h e n w e have:
(I)
X is b i r a t i o n a l l y surface
(II)
the
S such
image
of
nonsingular rational
that
~(S)
and
if X ( O x)
(I), w h i l e ,
whenever
then X belongs
PROPOSITION
1.3
Let X be
rational of the
exists
followin~
is a
fibre
o f e is a
X ( ~ x)
~ 0 and
K(X)
~:X + C such
and the general
threefolds
X
(II).
with
X is b i r a t i o n a l l y of
> 0 then
< O, q(X)
> O.
t h a t C is a fibre
equivalent
is a to o n e
X:
~2;
X = C x
(b)
X is a c o n i c
There
curve
Then types
(a)
(c)
e:X + AIb(X)
to f a m i l y
a threefold
a morphism
projective
surface.
to C x
mapping
a 0 and hO(sl2(~))
= O,
nonsingular
on a
or
the g e n e r a l
hO(sl2(~))-
there
bundle
surface.
to f a m i l y
Suppose
to a c o n i c
a O,
the A l b a n e s e curve
Furthermore belongs
equivalent
bundle
on a surface
S birationally
equivalent
pl ; exists
S is a D e l
a morphism
Pezzo
the anticanonical
surface sheaf
X ~ C such with -i ~S
that
Pic(S)
the ~eneric
= Z @enerated
Moreover
2 1 ~ ~S ~ 6.
fibre by
38
REMARK
1.4
of
previous
the
The
birational
2 ~S = 5 does
case
Proposition.
to C x p 2 .
construction
due
In
This
is
to E n r i q u e s
not
occur
in
(c)
2 if ~S = 5 t h e n
fact,
a consequence
(see
family
[E],
of
§8 a n d
X is
a classical
also
[Co],
p. 4 7 3 - 4 7 4 ) .
We about Ch.
summarize
conic
I in
here
bundles
the
case
without
on a surface
S = ~2.
properties
the
arguments
a standard
way
to
THEOREM
1.5
Let
(i)
The morphism
(2)
Let
~X be
f:X
Since
of
+ S be
some well
S.
They
all
questions
contained
case
f is
in
sheaf
is a l o c a l l y
free
There
a curve
C i__nn S w i t h
- for - if
every
f
-I
s is (s)
to
involve
[B], local
extended
in
surface.
Then
Then
sheaf
a quadratic
s e S\C
two
an ordinary is
in
we
have:
the
of
direct
rank
3 and
form.
at m o s t
ordinary
that:
s is a n o n s i n @ u l a r
i_~s i s o m o r p h i c - if
such
point
results
proved
c a n be
o n X.
by
points,
known
flat.
in ~(£)
double
[B]
bundle.
X is d e f i n e d exists
are
a nonsingular
a conic
canonical
6 = f,~l
ima@e
(3)
the
the
proof
isomorphic
the
point
fibre
of C then
distinct double
f-l(s)
is n o n s i n ~ u l a r ;
the
fibre
f
-i
(s)
lines;
point
to a d o u b l e
of
C then
line.
the
fibre
39
DEFINITION
1.6
discriminant is o r d i n a r y
The curve C of s t a t e m e n t
c u r v e of X.
M o r e o v e r we say that a conic b u n d l e
if the d i s c r i m i n a n t
nonsingular.
be the
the curve
GI(£)
f
-i
in GI(£)
whose
(s), s ~ C.
+ S induces
DEFINITION
1.7
equivalent
For m o r e d e t a i l s
f.e.
[M2].
follows. choose
of o r d e r
D
g i v e n by
@
principal
that
curve
restriction
polarization of PX'
can be o b t a i n e d > O.
see
as
We c a n
C such t h a t C £ 2D for
linear e q u i v a l e n c e )
whose
such
to m*0 w h e r e m:P x ÷ J(C)
such that q(S)
irreducible
("£" means
2 in Pic°(S)
Let
to the c o n i c
polarization
conic b u n d l e s
Let S be a s u r f a c e
some d i v i s o r
let
projection
and f u r t h e r p r o p e r t i e s
of o r d i n a r y
a nonsingular
and
the lines of
u:C + C.
of J(C)
and 8 is the c a n o n i c a l
on J(C).
Examples
are
PX a s s o c i a t e d
subvariety
PX has a p r i n c i p a l
is the i n c l u s i o n
in ~ ( 6 ) ,
closed points
etale covering
conic bundles.
0
is a l g e b r a i c a l l y
[B],
and
involution.
- T*).
Moreover
ordinary
of the lines
The P r y m v a r i e t y
X is the a b e l i a n
PX = Im(~j(~)
20
is i r r e d u c i b l e
T h e n the n a t u r a l
a double
T:C + C be the c a n o n i c a l
bundle
consider
be the G r a s s m a n n i a n
fibres
curve
Q
F r o m now on, we only Let GI(6)
1.2 is n a m e d
and an e l e m e n t
to Pic°(C)
is not
n
40
trivial.
If {h }~ is a s y s t e m of local
equations
of C,
then
the e q u a t i o n
~ox~+ 4+ 4 ~0 locally
defines
a conic
bundle
the d o u b l e in Pic O (C) .
in the p r o j e c t i v e X having
covering
scheme ~((gs(D)
C as d i s c r i m i n a n t
~:C ÷ C d o e s n ' t
split
curve. since
• £9S(n) @ C9S) Moreover ~ is not
zero
41
§2.
The C h o w group
Let
f:X + S
A 2(x)
of a conic
be a conic bundle.
the C h o w group of the c y c l e - c l a s s e s rational classes
equivalence,
by
algebraically
find a d e c o m p o s i t i o n Prym
variety
Aq(x)
First of all,
We denote
cq(x) q
modulo
the s u b g r o u p
to zero.
for the group
by
of c o d i m e n s i o n
¢ cq(x)
equivalent
PX and A2(S)
the r e s t r i c t i o n
bundle.
of the
Our p u r p o s e
A2(X)
in terms
is to
of the
• AI(s) .
choose
a very
~
ample
(1)~O X
sheaf on
is a very
S
such that
ample
sheaf on
~(f,~l~[) X([EGA],II,4.4.10) = ~(f,~l)
and
Therefore
nonsingular
divisor
system
f
I~
We say that be seen of
L
look at the e m b e d d i n g
-i
TX
Tx
we can
find an i r r e d u c i b l e
belonging
(i)~ ~ X
'
to the c o m p l e t e
linear
by use of B e r t i n i ' s
is a t a u t o l o g i c a l
in the sequel
X ÷ ~(f,~xl~i)
our a r g u m e n t s
divisor
of
X
do not d e p e n d
Theorem.
.
As it can
on the choice
(see 2.1.4).
2.1. Y = f-l(c) U = X\Y
Let
in this
i
be the d i s c r i m i n a n t
be the ruled
We d e n o t e U , by
C
by
surface
of the s i n g u l a r
s e c t i o n we w a n t T
to d e s c r i b e
the r e s t r i c t i o n
the i n c l u s i o n
T + U
curve of
of
Tx
and we set
X
and let
fibres.
Put
the group
A2(U)
to the open V = S\C
set
The
42 morphism
~ = f0i is g e n e r i c a l l y
we get the c o m m u t a t i v e
UT
where
change
~----[--->
LEMMA
2.1.1
equivalent one has
to a f i b r a t i o n
A 2(U T)
defined
f:U T + T
i be the i n v e r t i b l e
section
s.
The f u n c t i o n
is b i r a t i o n a l l y lines.
Moreover
sheaf on U T d e f i n e d
= dimHO(~-l(t) ,Lt)
Moreover
flat m o r p h i s m .
Therefore
rank
= h 0 (t,i)
(f,i®k(t))
(t,i(t)).
@ s,A I(T) .
Let
on T.
=
isomorphism
Proof.
is c o n s t a n t
by s(t)
in p r o j e c t i v e
= ~ * A 2(T)
hO(t,i)
diagram
we study the g r o u p A2(U) .
The p r o j e c t i o n
a natural
2 and
V
s:T ÷ U T is the s e c t i o n To b e g i n with,
of d e g r e e
U
>
T
base
finite
= d i m H O ~ I, ]pl(1))
i is flat on T since
f,i
is a locally
(see [HI,
by the
free
p. 288).
= 2
f is a sheaf
On the
and
43 other
hand
there
exists
a closed
embedding
% UT + ~(f,i), %
which
is just an i s o m o r p h i s m
of d i m e n s i o n projection
lines.
"." m e a n s
can write
%
5.5.4).
equivalent
H e n c e we have
= ~*Cq(T)
T h e n the
to a f i b r a t i o n
an i s o m o r p h i s m
~ ~ * c q - I ( T ) "s,AO(T),
intersection
of cycles.
(see
q = 1,2,
In p a r t i c u l a r
we
for e v e r y y ~ A 2 (UT):
y = %, f e + %, f ~-s,(T),
Since
III,
are b o t h
IV)
cq(u T)
where
%
(see lEGAl,
f is b i r a t i o n a l l y
in p r o j e c t i v e [C], exp.
three
as U T and ~ ( f , t )
%*
e e C2( T),
8 E C 1 (T).
%
f,f ~ = O and f,s, = id. we o b t a i n
%f,y = %f , ( A~* 8 - s , ( T ) ) h e n c e ~*~
£ A 2(UT).
= 8"[,s,(T) Therefore
= ~. s*~*~
It f o l l o w s
8 e A 1 (T),
= ~ E A 2(T) .
Then one has
A 2 ( U T ) -~ ~*A 2(T)(D~*A I(T) "s,AO(T) = ~*A 2(T)~)s,A I(T) .
q.e.d.
Consider ~U
now the m o r p h i s m : A 2 (V) @ AI(v)
+ A 2 (U)
44
defined
by ~u(a,b)
PROPOSITION
2.1.2.
Proof.
We have
written
in the
(a,b)
to s h o w form
all
elements
f*a + f * b - i , T ,
of A2(U)
can be
for some
since
£ A 2 (T) • A 1 (T) by L e m m a
(~,8)
f,f*~
= O and
Moreover
f,w
= n'f,
8 = n*8'
with
B' = f,Y
~*A2(U)
Recalling 2)
that
~U is s u r j e c t i v e .
a c y c l e ~* ~ y e A 2 (UT) , y e A 2 (U). We c a n w r i t e
= %, f ~ + s,~,
Then,
The morphism
E A 2 (V) @ A 1 (V). Take
~*y
= f*a + f * b . i , T .
that
f,s,
= id.,
IF],
2.2 ). T h e r e f o r e
(see
~ A 1 (V),
÷ f*A2(T)
o n e has
2.1.1.
f,n y = 8. we
find
so we get an e m b e d d i n g
S s,~* A 1 (V).
the c o m p o s i t i o n
is s u r j e c t i v e ,
the d i a g r a m
~*A2(T)
~ s,~*AI(v)
~,~
=
.2
(multiplication
by
N
~*A
is c o m m u t a t i v e
(')
A2(U)
with
(U)
exact
W*
> A 2 (U) + O
n~
row
and g i v e s
= ~,f~, A 2 ( T ) ~ ,~ s ,
, A 1 (V)
an i s o m o r p h i s m
= f*~,A2(T)~i,~*AI(v).
4B
Now we
compute cycle
i,~*AI(v).
take
the
i*f*n,x
~,~*
= "2 o n e has
i,(2x)
The group element
Since
= n*w,w*x'
= i,(i*f*w,x)
shown
so we
t o be o f t h e that
x E w*AI(v)
for such
i'f*
= 2x.
E ~*AI(v), = ~* a n d Then
in A 2 ( U ) :
= f*~,x.i,T.
is d i v i s i b l e ,
y ~ w*AI(v)
an element
all x = ~*x'
~ A 1 (T).
i*f*w,x
w*AI(v)
Thus we have
For
can assume
f o r m y = 2x,
all y ~ ~ * A I ( v )
every
x E ~*AI(v) .
there
exists
that
i,y = f * ~ , x - i , T .
Recalling
(-)
and p u t t i n g
b = ~,x we
are done. q.e.d.
As done
for the m o r p h i s m
=
such
(f*,i,~*)
: A 2 (S) • A 1 (S)
÷ A 2 (X)
that
~(a,b)
where
~U' w e c a n d e f i n e
i : T X ÷ X,
= f*a + i,w*b
~ = foi
= f*a + f * b - i , T x
: T x + S.
The morphism
# needs
48
not
to be
surjective
G(X)
plays
are
REMARKS
2.1.3
the
choice
embedding
i)
The
the
line
X ÷ ~(f,~
tautological
L'
map
that
#'
Im#
Proposition
=
2.1.2
Z generated
since
~,~*
suitable for it
all
=
(a,b)
L).
i on
To
see
corresponding e Pic(S),
The
In
find
following
= f*C
1
this,
It
is
(S)
of the
.2, w e
can
write
2(i,T~)
ImP'
c Im¢;
the
T x, b e
other
the
inclusion
the difficult as
to
in
• Z,
class
~ A 2 (S)
some
on
isomorphism
the
£ Z,
let
not
depend
gives
reasoning
by
m
not
consider
fact,
an
does
S which
to
and
(f*,i,~*) .
we
~ e C 1 (S),
follows
2.2.
G(X)
bundle
= ImP'
~ 1 ~ , ~ * C (X)
with
in s e c t i o n
quotient
i0
divisor
X +~(f,~xlO['),
verify
role
needed.
of
associated
cokernel
= A2(X)/Im#
an i m p o r t a n t
remarks
: the
so t h a t
• A 1 (S). similarly
As
cycle
i , T x.
= f*~
+ mi,T x
2#' (a,b) ImP'
Im#'
is
Then, for
= ¢(2a+b-~,mb)£Im~, a divisible
~ ImP.
group
47
2)
The
kernel
of # is a f i n i t e
in A2(S) 4 • A I ( s ) 2 . f,(f*a
+ f*b-i,T
f,(f
x)
In fact, = O.
b - i , T X)
b E A 1 (S)2,
Therefore
2(f*a
It f o l l o w s
2.2. conic
fibres. that
Xs
f
-I
e(s)
(s),
As
to
exists
= f*2a
+ f * 2 b - i , T x = f * 2 a = O.
Hence,
as b e f o r e
= 2 a . f , i , T x = 2 a - ~ , T x = 4a,
proved
in
[R] the g r o u p
the discriminant surface
a canonical
is t h e s i n g u l a r
for
the b l o w i n g - u p divisor
Then
a = O we f i n d
= f * 2 a - i , T x.
X a n d the r u l e d
There
such
E ker~
A2(S)4
AI(s) 2
to A I b ( S ) 4 • P i c ° ( S ) 2 , so it is f i n i t e .
We return
bundle
f,f
(a,b)
contained
so that
O = i*f*2a
is i s o m o r p h i c
Since
+ f * b ' i , T x)
is a ~ A 2 (S) 4.
suppose
group
= b . f , i , T x = b . ~ , T x = 2b.
f , ( f * 2 a - i , T x)
that
torsion
all s c C
of X along
a n d b y Y'
p':Y'
÷ C be the r e s t r i c t i o n
fibre
p'-l(s),
C of the
(C) o f s i n g u l a r
e : C ÷ Y fibre
b y e:X'
÷ X
by E the exceptional
transform of
-i
of t h e
We denote
the proper
s £ C,
Y = f
section
point
e(C),
curve
foe
is i s o m o r p h i c
of Y.
to Y'
Let The
to the b l o w i n g - u p
48
of
X
in its
s
singular
point.
p'
Furthermore
factorizes
p,~
through
p
!
Y'
-"
C
C
where
p
double
etale
are
is a f i b r a t i o n covering
nonsingular
intersection projection
with
in p r o j e c t i v e
of
C
Moreover
transversal
can be i d e n t i f i e d
p.
Thus
we h a v e
lines
and
the
is the
the d i v i s o r s
intersection.
with
~
curve
a commutative
E,Y'
Their C via
the
diagram
E
C
X'
h",.
y,
/<.3 £
%
C
C
where
the
upper
is a s e c t i o n
square
X
r
,
S
is c a r t e s i a n
of the m o r p h i s m
p
,
and i.e.
the poh
inclusion = id
h
49 Write
U = XkY,
V = S\C and let JU :U + X,
rV:V ÷ S be
the inclusions.
LEMMA
2.2.1a)
The sequence (£0j),
j~
AI(y ')
> A2(X)
> A2(U)
÷ O
is exact. b)
r~:AI(s)
Proo~a)
Look
the subgroup
+ AI(v)
is an isomorphism.
at K = kerj~
as subgroup
K O = (e0j),-IK
CI(y ')
(eoj), ->
is exact and the morphism (see [B], 0.1.2)
of CI(y').
C2 (X)
we get the exactness
o
Since the sequence
> C2(U)
+ O
of
j~ > A2(X)
Then it is sufficient sequence
j*U
and consider
* : A 2 (X) + A 2 (U) is surjective 3U
(eoj), K
of C2(X)
to show that K
~ A2(U)+
o
= AI(y').
O.
The last
gives AI(y ') c K O.
To prove the converse, the classes
of divisors
let Picn(y ') be the group of
numerically
equivalent
to zero.
The
50
quotient
group
Num(Y')
by C o and F where Choose shows
= Pic(Y')/Picn(y
F is a f i b r e of
an ample divisor that
D o n S.
the d i v i s o r s
n o w y E K o. as e l e m e n t
Then
(e0j),y
of A 2 ( X ) .
~ E Pic(X).
The
so t h e c l a i m
implies
and algebraic
a direct
to t h e
over
~.
equivalent
(e0j),y-~
formula
calculation
sheaves
Num(Y')
is n u m e r i c a l l y
Therefore
y.~
+ C and C O a section.
to
equivalence
to z e r o
coincide
zero
= O for all
gives
y.(e0j)*~
= O,
= O for all ~ E N u m ( Y ' ) .
equivalent
Suppose
in Y'
Since
on the ruled
That
is
numerical
surface
Y',
y E A 1 (Y').
it f o l l o w s b)
Then
generate
projection
y is n u m e r i c a l l y
p:Y'
associated
(eoj)*CQx(f*D) , (eoj)*~X(1)
') is g e n e r a t e d
Set K ° = k e r
r~ and c o n s i d e r
the
exact
diagram
(see
[B], 0.1.2) r,
c°(c)
K
o
cO(c)
cO(c)
c a n be w r i t t e n
numerically
K
o
=
r* V
cycle).
>
C1
(V)
class
~C
to z e r o Then
O
This
as c y c l e
÷ O
all
elements
a ~ Z.
y E K ° so t h a t
to z e r o .
÷
3 > A I(V)
f o r m y = a] C,
Suppose
equivalent
r~
by t h e
i n the
equivalent
is an e f f e c t i v e
--
A I(S)
= Z generated
r,y = aR C in C I ( s ) .
(S)
3
r, - - >
Since
algebraically
C1
>
It f o l l o w s
a I c is
implies
a l C to b e
of S, h e n c e
y = O in c O ( c ) .
(O), a n d w e a r e done. q .e .d.
Y in
This
a = O(~ C means
51
Next
define
~A
and
=
let G(X)
introduced
the c y l i n d e r
e,j,p*
: J ( C~ )
= A2(X)/Im~
in s e c t i o n
~:
J(C)
map
+
A 2 (X)
b e the
2.1.
cokernel
Denote
of the m a p
by
÷ G(X) q
the c o m p o s i t i o n
PROPOSITION cylinder
Proof. there y-8
2.2.2
map
exists
classes
a cycle
A2(X)
~:J(C)
closure
belongs
to A2(U) #u:A2(V)
for
>
+ G(X)
all
{j~:A2(X)
by L e m m a
8 belonging
simplicity, in the
that
8 E ker
~ and J(C)
Zariski
morphism
to p r o v e
a cycle
x £ J(C)
y £ A 2 (X)
projection
G(X) .
induced
by the
is s u r j e c t i v e .
Therefore
of
For
the
The m o r p h i s m
It s u f f i c e s
¢Im#.
where
with
2.2.1
÷ A2(U)} all
via
set u = JU"
We
p*,
of u*y
in X and
as y ~ A 2 (X) . ~ AI(v)
÷ A2(U)
so that
~ = u*y
6 = y-e. Then
that
can w r i t e
form y = ~ + 6 where
the
(see
such
elements
to I m ( e 0 j ) , ,
= A 1 (Y')
y ~ A 2 (X)
cycles
The
of G(X)
are
is ~ = e , j , p * x , ~ is s u r j e c t i v e .
all c y c l e s E C2( X) cycle
surjectivity
2.1.2)
that
is the
u*~
= u*y
of the
gives
u*~ = f*a + f * b - i , T
with the
(a,b)
E A 2 (V) ~ A 1 (V), T = u*T x.
commutative
square,
q = 1,2,
N o w put v = r V and ccnsider
52
v*
A q (S)
Aq(v)
u* A q (X)
The
v*,
morphisms
hence Thus
we we
, Aq(u)
find
u*
are
surjective
a = v*a
, b = v*b
f v*a
+ f v*b-i,T
for
([B],
O.l.2(ii)),
some
(a,b)
e A 2 (S) C A 1 ( S)
have
u*a
=
u*a
= u*f
,
then
that
a + u*f
,
is
~-~(a,b)
By
b.u*i,T x
use
of
the
E ker{j~:C2(X)
exact
sequence
(see
÷ C2(U)}
again
[B],
0.1.2)
U* C1 ly, ~
we
find
in
C 2(x)
(£oj), )
:
C 2 (X)
~
C 2 (U)
÷ O
53
= ~(a,b)
+
= #(a,b)
+ 6 +
,
(eoj),~
~ c
C 1
Y-B
~ Imp.
(Y')
Hence
Putting
8 = 6 +
j~8 = O
and
(eoj),~
S E A 2 (X)
(e0j),~
one has so that
,
Moreover
8 e ker{j~:A2(X)
+ A2(U)}
as r e q u i r e d . q.e.d. N o w we p o i n t o u t some p r o p e r t i e s ~A
' ~
CLAIM
Proof.
The n o t a t i o n s
i.
are as in s e c t i o n
For all
e E J(C)
e,j,p*2~
= f * e',
Consider
A 2(S) r~
A2(V)
The h y p o t h e s i s
such that
#
~
~ j~
~ AI(v)
~(a)
diagram
, A 2(X)
OU ,
A2(U)
= O implies
2.1.
~(e)
~'e A 2 (S)
the c o m m u t a t i v e
(9 A I(S)
of the m o r p h i s m s
= O
one has
54
e,j,p*~
for as
some in
the
(a,b)
e A 2 (S)
proof
of
j~(f
a +
3.~ i , .= 1 , 3. ~
Since last
= f a + f b-i,T x
equality
is
Proposition
f b . i , T x)
,. we
the
same
ker~u
Hence
r~
e,j,p*2~
has
.
3 u*' I,T x = i,T
so
that
the
.
as
in
Remark
2.1.3,2)
we
find
c A2(V) 4 @ AI(v)2
in p a r t i c u l a r
: A 1 (s)
= O
one
reasoning
to
= O
argument
Therefore 2.2.2
see
equivalent
~u(r~(a,b))
With
@ A 1 (S)
÷ A 1 (V)
r~(2b) is
an
= O
which
isomorphism
implies (see
2b
2.2.1b)).
= O
since
Thus
= f 2a. q .e.d.
Let
now
T*:J(C)
÷ J(C)
be
the
canonical
involution
rU
induced
by
the
double
etale
covering
:C ÷ C
Then
we
55
have
CLAIM
2.
The
equality
p,j*e*~A(e)
holds,
for
Proof.
all
e ~ J(C)
It r u n s
is p r o v e d
= T*e-e
as w e l l
in the
case
as
in
[B],
3.1.4
where
the
assert
S = ~2
D Recall
the
definition
associated
to
PX
the
cylinder
by
Pn
of
denote
LEMMA the
2.2.3
X
the
Let
discriminant
of
Let
P:Px
map
~A
set
9:C
Prym
÷ A2(X)
curve
C
be
be .
the Then
of
one
restriction
+ A 2 (X)
order
double
PX = I m ( ~ j ( ~ ) - T * ) the
= e,j,p*:J(C)
of points
÷ C
variety
n
of
etale
,
to
and
PX
covering
of
has
ker (~j ~ -T*)=v*J (C) (C)
Proof. since
The T*r*a
To
prove
relation = ~*a the
~*J(C) for
all
converse,
c ker(]ij ~ -T*)
(c)
is c l e a r
a ~ J(C) take
a =Cg~(D)
~ J(C)
such
that
56
a = T*a
,
that
We can choose has positive induced of
by
is
a divisor
Put
B
of
D H 9*(B-A)
C
that
. is
on
Let
C i*
i*E = E
Moreover
linear
such be
E = div(s)
Then we have
divisor
("5" m e a n s
A
dimension. T
i*
D E T*D
that
V = HO(c,~(D+9*A))
the e n d o m o r p h i s m
where ,
equivalence).
s
hence
of
V
is an e i g e n v e c t o r E = ~*B
E H D + ~*A
for s o m e
It f o l l o w s
a c ~*J(C) q.e.d.
The
following
for the g r o u p
PROPOSITION ~x:Px
commutative
gives
the r e q u i r e d
decomposition
A2(X)
2.2.4
÷ G(X)
Proposition
i)
whose
There kernel
the d i a g r a m
a surjective
is c o n t a i n e d
with
0 ÷ I m ~ ÷ A2(X) 01
exists
exact
q~
J /
in
P2
morphism which
makes
row
G(X)
xO(.2)
÷ 0
=
x
PX
ii)
A2(X)
iii)
p ( k e r 2 ~ x)
iv)
the m o r p h i s m
Proof.
i)
= Im# + p(Px ) ; ~ Im~
We have
n p(Px ) ;
~+p:A2(S)@AI(s)@Px
to s h o w
that
there
÷ A2(X)
exists
is an i s o g e n y .
a surjective
57 morphism
with
ker~x
PX
m
c P2
such
CA
, J(C)
that
q
~x
G (X)
PX
the
Putting
~ = q0~A
Suppose
~ = ker(ij(~)-~*)
a = T*a
6 e J(C)
by
also
3.1.5),
[B],
Since
f r.B Assume a'
e
,
Lemma
~(a)
=
that
2.2.3.
it
is
sufficient
to p r o v e
j*e*f
, 2 ker
a = ~*B Moreover
~ c ker(]ij
for
~ -T*)
(c)
a suitable
e.j.p*~*
=
f r.
]" m e a n s
the
(see
hence
[f r . 6 ]
e Im~ now
so
,
it
~(~)
= O
, where
follows
"[
~(a)
, ~ ~ J(C)
= O
in
Claims
class
= p*~*r*
,
we
= p.j*e*(f
find
e')
in G(X) .
G(X) 1,2
A 2 (S)
p.j*e*(e.j.p*2e)
AS
,
relations
ker
some
diagram
, A 2 (X)
]ij (~) - T *
commutes.
the
= 2(T*~-e)
give,
for
58
p,p*~*r*~'
Since
r*:A2(S)
2T*e
= 2e
ii),
iii).
iv).
,
= 2(T*a-~)
÷ Al(c)
that
is
They
is the
immediately
¢+p
by
i).
is s u r j e c t i v e
(a,b,c)
c ker(¢+p)
, so t h a t
Moreover
ker2~x
is a f i n i t e
i) ,
t h e n we o b t a i n
2~ E k e r { ~ j ( C~) -T*)
follow
The m o r p h i s m
zero map,
iii)
by ii).
implies
group
Take
¢(a,b),p(c)ep(ker2¢x).
of o r d e r
d
(d <_ 4)
by
therefore
¢(da,db)
= p(dc)
Since
ker # c A 2 ( S ) 4
which
is a f i n i t e
=Alb(S)tors c e Pd2
Thus
@ AI(s)2
group
proved
= O
we
in v i e w
in [R],
ker(~+p)
find
(a,b)
of the
~ A 2 (S)4d @ A 1 (S)2d
isomorphism
while
p(dc)
is a f i n i t e
group.
= 0
A2(S)tors = implies
q.e.d.
REMARK over
2.2.5.
the
complex
conditions
one has
pg(S)
= O
A2(S)
= O
AI(s)
= O
A2(S)
isomorphism
as .
= pg(S)
the
and
X + S
numbers.
q(S)
type,
mapping
Suppose
Then whenever = O
isomorphism
S
is n o t
+ AIb(S) q(S)
= O
to be a c o n i c
,
and ~x:Px
of g e n e r a l is i n j e c t i v e Moreover
S
= G(X)
= A2(X)
defined
satisfies
the
is n o t of g e n e r a l
= A2(X) type, (see q(S)
R e a s o n i n g . as in P r o p o s i t i o n ~x:Px
S
bundle
In fact,
if
the A l b a n e s e [B-K-L]), = O
2.2.4,
also
so t h a t implies
we get
the
59
§3.
The a l g e b r a i c
representative
of A2(X).
We show in this s e c t i o n that an a n a l o g o u s Proposition
2.2.4 holds
A x of A2(X).
true for the a l g e b r a i c
To b e g i n with, we recall
Here V is any n o n s i n g u l a r
DEFINITION
3.1
representative
some d e f i n i t i o n s .
threefold.
Let W be a n o n s i n g u l a r
say that a map
result to
(of sets) 8 u:W + A2(V)
a l g e b r a i c variety. is a l g e b r a i c
We
if there
exists a cycle Z E C2(V x W) such that for all t ( W,
u(t)
DEFINITION
= Z-({t} xV)
3.2
Let B be an a b e l i a n variety.
We say that a
m o r p h i s m of groups ~: A 2 (V) ~ B is r e g u l a r if for e v e r y a l g e b r a i c m a p u:W ÷ A2(V)
the c o m p o s i t i o n
DEFINITION
Let av:A2(V)
3.3
a b e l i a n variety. representative
Sou is a m o r p h i s m of v a r i e t i e s .
+ A v be a r e g u l a r morphism,
We say that the pair
of A2(V)
Av
(Av,a v) is the a l g e b r a i c
if, for all r e g u l a r m o r p h i s m s
8:A2(V)
there exists one and only one m o r p h i s m h:A V ÷ B of a b e l i a n varieties w h i c h m a k e s the f o l l o w i n g d i a g r a m A 2 (V)
av
B
commute.
>
Av
÷ B,
60 REMARK 3.4
For every nonsingular variety of dimension n z 2,
analogous definitions are given for the groups Aq(v), q ~ n. In particular, (resp. Pic°(V)) AI(v)) .
it is not diffi~=ult to verify that AIb(V) is the algebraic representative of An(v)
Moreover if the pair
(resp.
(Av,a v) exists then it is unique
up to isomorphisms and the morphism av is always surjective (see [B], 3.2.4). The following Proposition states the existence of the algebraic representative of A2(V)
for a large class of three-
folds, including conic bundles.
PROPOSITION 3.5
Let Y,V be nonsin~ular threefolds and let
#:Y -- --> V be a rational map. al~ebraic representative
Suppose that there exists the
(Ay,ay) o f A2(y).
Then there exists
also that one of A2(V).
Proof.
We get a commutative diagram
R
f Y
--> V
where R is a nonsingular threefold,
0 is a sequence of blowing-
ups along points or nonsingular curves, and # is a morphism.
61
Since
the b l o w i n g u p
A2(y)
nor its a l g e b r a i c
be a sequence
along points
does not change
representative,
of b l o w i n g u p s
along
Therefore
one can e a s i l y
prove
A R = Ay •
H J(C i) is the a l g e b r a i c i
we
neither
can suppose
nonsingular
curves
that the d i r e c t
c to
C i-
sum
representative
of the
group A2(R). Now, variety.
let 8:A2(V) It is easy
is regular.
Then,
representative,
÷ B be a r e g u l a r
morphism,
80~, :A 2 (R) ÷ B
to see that the m o r p h i s m
by the u n i v e r s a l
there exists
property
a morphism
B an a b e l i a n
of the a l g e b r a i c
g such
that the
diagram
A2(R)
> A 2 (V)
aR g >
AR
co~utes.
Since
the m o r p h i s m
B
~, is surjective,
h e n c e one has
d i m Im 8 = d i m . Im g < d i m A R = M for all pair
(B,8) •
equivalent
representative
of A 2(V),
to the e x i s t e n c e as p r o v e d
in [S],
of the a l g e b r a i c
This
is
2.2. q .e .d.
COROLLARY particular
3.6.
Suppose
V to be a u n i r u l e d
a conic bundle).
representative
A v o_ff A2(V) .
Then
there
threefold
exists
(i__nn
the a l ~ e b r a i c
62 Proof.
As V is u n i r u l e d
there exist a n o n s i n g u l a r
Y and a rational map y x ~ l a b e l i a n v a r i e t y AIb(Y) of A2(y x ~i)
_ _> V.
@ Pic°(Y)
via the c a n o n i c a l
surface
On the o t h e r hand the
is the a l g e b r a i c r e p r e s e n t a t i v e i s o m o r p h i s m A 2 ( y M P 1 ) -~ A 2 (y) @A 1 (Y).
T h e n the c l a i m follows by the p r e v i o u s
Proposition. q.e.d.
We go back to the case of a conic bundle the n o t a t i o n s
f:X ÷ S.
With
as in s e c t i o n 2.2, look at the c o m m u t a t i v e
diagram
X
,
Y
g
P
f
>
X
-->
S
>
C
- - >
C
and c o n s i d e r a g a i n the m o r p h i s m #:A2(S) the i s o g e n y ~x:Px ÷ G(X) recall
@ AI(s)
÷ A2(X),
and
= A 2 ( X ) / I m # (see s e c t i o n 2.1).
We
that ker ~X is c o n t a i n e d in P2' so that there exists
an i s o g e n y ~x:G(X)
÷ PX such that ~X 0 #X = -2 is the m u l t i -
p l i c a t i o n by 2 in PX" We need the f o l l o w i n g t e c h n i c a l
LEMMA
3.7(i)
Let q:A2(X)
+ G(X)
Then
the c o m p o s i t i o n ~X 0 q :A2(X)
result
be the c a n o n i c a l p r o ~ e c t i o n . ÷ PX is a regular m o r p h i s m .
63
(*)
(ii) Then
Let u:W + A2(S) #ou:W ÷ A2(X)
Proof: cycle
(i)
•
AI(s)
composition
map
is also.
Let u : W + A2(X)
z ¢ C 2 (XxW)
be an a l @ e b r a i c
be an a l g e b r a i c
and put u = qou.
~X ou is a m o r p h i s m
We have
of a l g e b r a i c
map defined
by a
to s h o w that the varieties•
Consider
the d i a g r a m
WxY '
> WxX'
>
WxX
% WxC
where
". ' " means
defined
by the cycle
d.({t}xC),
the c o m m u t a t i v i t y
We m e a n maps,
d
=
.
Take
p~j'*
the a l g e b r a i c
e'*z
~
t ~ W • V i a the i s o m o r p h i s m
ition v:W ~ J(C)
(*)
(i%,-)
is just a m o r p h i s m
C 1
(WxC),
A 1 (C) ~
m a p v:W + A 1 (C) ~ that is v(t)
= J(C),
of a l g e b r a i c
the
that u =
i = 1,2.
(Ul,U 2) w i t h
ui:W ÷ Ai(s)
compos-
varieties.
of the diagram:
algebraic
=
Moreover
64
XxW <
a
Xx {t}
e'oj/
/
Y'x W
C xW
b
<
Y'x{t}
c
<
eoj
Cx{t}
gives j*e*(z.({t}xX)) Put x = j'*e'*z.
= j*e*a*z
Then we have
= b*j'*e'*z
(see [F],2.2
),
p , b * x = c*p~x = p~x. ({t}xC)
that is (*)
Consider
p,j*e*(z.({t}xX))
= p~j'*e'*z.({t}xC)
n o w the d i a g r a m
W
]Ij{ ~ - ~ * /
(=v(t)) .
85
where the
u = qou,
composition
Proposition
~ =
(Ij(~)
~ = qo~A
2.2.2),
S t £ J(C)
such
for
that
u(t)
- T*)0
v and #xO~x
is a s u r j e c t i v e all
t
~ W there
= ~(st),
Since
morphism exist
that
=.2.
(see
some
is
[z-({t}xX) ] = [e,j,p*st] ,
where
"[ • ]" m e a n s
the c l a s s
z-({t}xX)
for some
(s,~)
v(t)
Claim
= e,j,p s t +
E A 2 (S) • A 1 (S).
Then we
find
~(s,8)
Hence
b y use o f
(*):
2.2 g i v e s
p,j*e*(c,j,p*ut)
a standard
Proposition
Imp.
= p , j * E * ( e , j , p * s t) + p , j * e * ~ ( s , 8 ) .
2 of Section
Moreover
modulo
= T*s t - s t
computation
shows
.
(see the p r o o f
2.2.4) :
2(~j(~)
- ~*) ( p , j * e * ~ ( u , 8 ) )
= O.
of
66
Therefore
we have
2v(t)
Thus
= 2(Ij(~)
the e q u a l i t i e s
4u = - ~ x O 2 V ,
= 4(T*u t - ut ) .
= ~ ( u t ) = #xO(Ij(~)
- T*)(s t) imply
hence
-~xO4U
Since v, OxOOx plication
u(t)
- T*)v(t)
= #xOd#xO2V.
=.2 a r e
morphism~of
by 4,2 are isogenies,
varieties
and
the
multi-
also ~xOU is a m o r p h i s m
of
varieties.
(ii)
Let u:W + A2(S)
x E C 2 (SxW),
the cycles (x. ({t}xS), that z=
e AI(s)
y ~ C 1 (SxW),
y.({t}xS)).
map d e f i n e d
that is u(t)
A straightforward
~ou is the a l g e b r a i c f'*x + f'*y.(TxXW)
be the a l g e b r a i c
map d e f i n e d
~ C 2 (XxW) w h e r e
by
=
computation
shows
by the cycle f' =
(f,id W) . q .e .d.
With can prove
the same
notations
as in P r o p o s i t i o n
2.2.4,
we
the f o l l o w i n g
PROPOSITION
3.8
There
dia@ram with exact
row
exist morphisms
hl, h 2 and a c o m m u t a t i v e
67 h AIb(S)
• Pi~
(S)
1
h2 AX
-
> PX ÷ 0
aXo P
Px Moreover
(i)
A X = Im h I + aXo p (Px) ;
(ii)
aXo p(P2 ) ~ Im h I n aXo p (Px) ;
(iii) Suppose
the surface
S has i r r e g u l a r i t y
h 2 : A x ~ PX is an abelian v a r i e t y
(iv)
q(S)
= O.
Then
isomorphism.
In the complex case the m o r p h i s m h I + ax0P: AIb(S)
Proof.
• Pic°(S)
By use of Lemma
algebraic
• PX ÷ AX is an isogeny.
3.7 the universal
representatives
as, a x gives
h I, h 2 such that the following
property
of the
a b e l i a n v a r i e t y morphisms
diagram q
0
>
ker ~
> A2(S)
• AI(s)
~-~--> A2(X)
aSI
h2
h1
Note that a S
1 A 1 (S) ~ Pic°(S) as: The exactness while
=
> Ax
2 (as,a Si ) where
are the algebraic
the exactness
a s2 :A2(S)
>Px+O
÷ AIb(S),
representative
of the first row is p r o v e d
÷ O
~x
lax
A l b ( S ) O P i c ° (S)
commutes.
> G(X)
maps.
in P r o p o s i t i o n
of the second one is easily
achieved
2.2.4(i), by
68
taking into a c c o u n t that the m o r p h i s m s Now
(i) , (ii) , (iii)
immediately
be p r o v e d in S e c t i o n 4, 4.1.2,
a S , a x are surjective.
follow, w h i l e
(iv) will
4.2.5. q .e .d.
§4. The classical Through
intermediate
J a c o b i a n of a conic bundle.
this section we work over the c o m p l e x
In the first part we recall some d e f i n i t i o n s general statements. bundles
and give some
In the second one we go back to conic
and we get a d e c o m p o s i t i o n of the i n t e r m e d i a t e
Jacobian
4.1.
(up to isogenies).
Let
V
be a n o n s i n g u l a r
the c o m p l e x field
~
= @ HP'q(v) p+q=3 is the c o m p l e x torus
T2(V)
The
2 th G r i f f i t h s
= HI,2(V ) ~ HO'3(V)/poi
p:H3(V,~)
i:H3(V,~)
+ HI'2(V)
÷ H3(V,~)
8 HO'3(V)
the natural
in [G])
w : A 2(v) + T 2(v)
J a c o b i a n of
V
H3(V,~)
is the p r o j e c t i o n
inclusion.
Now we define the Well m o r p h i s m homomorphism
t h r e e f o l d d e f i n e d over
C o n s i d e r the Hodge d e c o m p o s i t o n
H3(V,~)
where
field.
(called A b e l - J a c o b i
and
69
Fix closed 3-forms of the
#'s
give a basis for
be a basis for ~N
#l,...,~N
H3(V,Z)
= ~N/F
H 2'I (V) S H3'O(v)
and let
generated by the vectors
T2(V)
such that the cohomology classes
C
The image
3-chain on w(A2(V))
J(V)
HI'2(V)/(H3(V,Z) We say that
Then
and the Weil mapping is given by
is a
subvariety
{yj}j
be the subgroup of
{([¥jj %1' .... I #N)}J 7j
x = ~C ~ (I ~i' .... I #N) C C where
F
Let
of
F ,
V .
under the morphism
T2(V)
n HI'2(V))
J(V)
modulo
,
w
is an abelian
which is contained in (see [G] n.2
is the intermediate
and ILl, n.3).
Jacobian of
V
Moreover the Weil morphism is re@ular in the sense of definition 3.2 (see IS], n.5). Therefore, algebraic representative then a (surjective)
w
~ J (V)
of the group
A2(V)
,
abelian variety morphism is defined such
that the diagram
A 2 (V)
(Av,a V)
if there exists the
70 REMARK
4.1.i.
n a 2
analogous
w :Aq(V) q with
For every nonsingular definitions
÷ Tq(v),
PROPOSITION
In particular
w
mapping
An(V)
+ Alb(V)
,
isomorphism
AI(v)
= Pic°(V)
let
4.1.2.
~:Y-~V
Let
Y,V
be a rational
algebraic
representative~
isogenous
to
~
isogenous
Av
Proof.
to
Prop.
via
(essentially
The existence 3.5.
of
~
to verify
av0~ , :A 2 (Y) ÷ A v
of
av,a Y
A 2(V) av I
AV
J(Y)
the to be
Then there exists
o_~f A2(V)
see
to be a morphism.
and
J(V)
[M-B],
i_~s
5.3).
in section
Moreover
3,
note Hence
~ A 2(Y) I
aY
it is not
ay0~*: A 2 (V) + ~
that the morphisms
the commutative
4"
and
again as in the proof of Proposition
are regular:
~*
is
exists
and assume hy
wI
threefolds
there
has been proved
finite morphism. gives
while
hV
(Av,a v)
Reasoning
generically
Suppose
due to Murre-Block,
3.5 we can assume difficult
map.
via the morphism
representative
coincides
n
be nonsingular
o_~f A2(y)
the algebraic
of dimension
give Weil morphisms
q ~ n
the Albanese
the canonical
variety
for this
that
the universal
~
ay[
Ay
4, 4,
is a
property
squares
A 2(Y)
,
~ A2 (V) [ av
71
Thus we get
the c o m m u t a t i v e
Ay hy
I
is a f i n i t e
ker hv that
is
~*a
group.
,
that
is
ker~*
c ker(~,o~*) ~
na
If
Since
o n e has
n c~
d = deg
it s u f f i c e s
E k e r hy
by hypothesis,
by
l
hy
#* >j(y)
~ J(V)
is s u r j e c t i v e ,
hv
' Ay
Ih v
J(Y) AS
~*
Av
~
diagram
n~*(a)
£ ker
,
~*
that
the k e r n e l
a c ker
hv
k e r hy
is a f i n i t e
= ~*(na)
, hyO~*(a)
= O
On the o t h e r
moreover
It f o l l o w s
to s h o w
~,o~*
nda =
0
= O
group
for s o m e hand
is the m u l t i p l i c a t i o n ,
so w e
are done. q.e.d.
L e t us c o n s i d e r particular
conic
PROPOSITION
the c a s e
of uniruled
threefolds
(in
bundles).
4.1.3.
Let
V
be a uniruled
threefold.
w e have: i)
J(V) Av
ii)
is i s o g e n o u s of
The Weil
to the a l g e b r a i c
representative
w : A 2(v)
is s u r j e c t i v e .
A 2 (V) morphism
+ T 2 (V)
Then
72
iii)
J(V)
is equal
to
Hl'2(V)/p0i
principally
polarized
Proof.
It is a consequence
i)
fact, as
V
is uniruled,
~:Yx]? 1 -÷ V, Y AIb(Y)
~ Pic°(Y)
w : A 2 ( y M P I) + T2(yxP I) to be a morphism. there exists
~*
images
In
map Then of
A2(yxP I) = A2(y)~AI(y)
formulas,
we get an i s o m o r p h i s m
therefore
the Weil m o r p h i s m
As usual,
Since the Well m o r p h i s m s
we can assume
are regular
square
~ A 2 (V)
wl T 2 (Yx~ 1 )
surface.
representative
is surjective.
a commutative
A 2 (Yx~ 1 )
,
4.1.2.
a rational
isomorphism
By use of Kunneth
T2(yx~ I) = AIb(Y)
The direct
projective
I!
ii)
of P r o p o s i t i o n
is the algebraic
via the canonical
and it is a
variety.
there exists
nonsingular
• Pic°(Y)
A2(yxp I)
abelian
H3(V,~)
lw #*
~,
~ T 2 (V)
are surjective,
so that
w : A 2(v)
÷ T 2(v)
is also. iii)
Recalling
we get by ii) :
that
HO'3(V)
= (0)
as
V
is uniruled,
73
J(V)
Moreover
= H l ' 2 ( v ) / p 0 i H 3(V,~)
= T 2(v)
J(V) has a canonicalprincipalpolarization 8.
ing h e r m i t i a n
The correspond-
form is d e f i n e d by
= 2i I ~^8_
H(~,8)
,
~,8
~ H I '2(V)
V The
fact that the p o l a r i z a t i o n
of P o i n c a r e d u a l i t y
(see also
is p r i n c i p a l [B], 0.2
and
is a c o n s e q u e n c e IT], Ch.I). q.e.d.
Through
4.2, bundle
this s e c t i o n we r e t u r n to c o n s i d e r
f:X + S
Y'
Look again at the c o m m u t a t i v e
~
,
a conic
diagram
X'
x
C
introduced
r
,
S
at the b e g i n n i n g
genus of the d i s c r i m i n a n t singular E:X'
÷ X
fibres,
curve
e:C + Y
the b l o w i n g
of s e c t i o n
up of
C, Y
2.2.
along
g
the s u r f a c e
the c a n o n i c a l X
Let
be the of
section,
e(C),
Y'
the s t r i c t
74
transform
of
Y
the e m b e d d i n g
As
of the
in s e c t i o n tautological
2.1,
let
divisor
i:T x + X Tx
,
be
and p u t
~ = foi:T x + S . We need canonical
some preparatory
involution
+
4.2.1i)
HI(C,~)
Let
T "C + C
be
the
and define
~
HI(C,•)
LEMMA
results.
of
= {y e HI(C,~.),¥TT*y
There
the
exists
= O}
a canonical
homology
basis
for
form
{~i ..... ° g ' O g + l ..... ° 2 g ' T ( ° l ) ..... T(Og_l) , T ( O g + l ).... ,T(O2g_l)}
such
that
Moreover
Og the
intersection
O
-I
is h o m o l o g o u s
I
and
° 2 g = T(O2g)
is
I
g
i
0
~-'r L
The
matrix
~(Og)
0
g
0
ii)
to
ZZ-module
I
g-1
g-1
0
H I (C,ZZ)
is @ e n e r a t e d
by
the s e t o f
cycles {Oi - T(Oi)}i
'
i = 1 ..... 2g-l,
i ~ g
75 Proof.
i)
See f.e.
ii)
[Fa],
It follows
II.
by i) since
H I(C,~)
= ker(]i+T*) q.e.d.
For any an element Ch. 0,4,
d-dimensional of
H2d_nl(V,~)
we denote by
section cycle.
~t:(f
the inverse
Poincare
LEMMA
Proof.
let
p*,f
,~*
the intermaps
,
are d e f i n e d
via
duality.
4.2.2
(Beauville)
for all
See [B],
The equality
= -a- (b-T'b)
a,b c HI(C,Z)
(or
be
As in [G],
now the topological
+ H3(X,~)
(e, 8)
: HI(S,~ ) $ H3(S,~ ) ÷ H3(X,2Z ) ,
images
~t(a) "~t(b)
holds,
,
x H2d_n2(V,~)
: HI(C,~)
,i,~*)
V
~'8 c H2d_nl_n2(V,~)
Consider
~t:e,j,p*
where
variety
a,b c HI(C,~))
2.3. D
76 The maps
~t' ~t
with rational
induce morphisms
coefficients,
which
of
homology
we d e n o t e
again
groups
by _
~t' ~t
Let
LEMMA
4.2.3(i)
(ii)
H3(X,~)
Proof.
(i)
~t
be the restriction
The morphisms
#t' ~t
Take
(a,b)
~t(a,b)-7
Beauville
~t
to
HI(C,~)
are injective
on
= ~t(HI(S,~)~H3(S,~))~t(HI(C,~))
# (O,O)
in
HI(S,~)eH3(S,Z)
It is easy to see that there exists such that
of
~ 0
formula
Then
gives
a cycle
7 E H3(X,~)
~t
ker~t
is injective. +~ c HI(C,~ ) Moreover
+~
H I(C,~) (ii)
n H~(C,~)
= (O)
To begin with,
(*)
Since
Therefore
is torsion
that for all cycles one has
free,
= (0)
it is sufficient
e ¢ Hi(C,~),
to show
¥ e ~t(HI(S,~)eH3(S,~))
~t(~) "y = O .
We are going to compute *
(a,b)
= (O)
we prove
#t (HI (S ,@) eH3 (S ,~) ) n ~t(HI(C,~))
H3(X,~)
~t(b)'f
kerSt
the intersections *
b'i,T x
where
e HI(S,~)SH3(S,~)
~t(e)'f
~ = f a + f b.i,T x , We have
a = f,(~t(e)"f
a) = f,~t(e)-a
~t(e)'f
a ,
77 On the o t h e r support
hand
~t(e)
is c o n t a i n e d
morphism
f.
= e,j,p*e
in a finite
Therefore
is a cycle whose
number
of fibres
= 0
so that
f,~t(e)
,
of the
~t (~) " f a = O , Moreover,
after
recalling
get from p r o j e c t i o n
~t(~)-f
formula
b =
that
j*e*f
= p*~*r*
the f o l l o w i n g
e,j,p*a.f
b
,
we
equalities
= e,j,(p*~-j*e*f
b) =
= e,j,p*(e'~*r*b)
_
~
+
By h y p o t h e s i s
e e HI(C,@)
construction.
Then
Propositions
d i m J(X)
,
e.~*r*b
3.~ and 4.1.3
< dim(Alb(S)
while
9*r*b
= 0
and
E HI(C,~)
(*)
by
is proved.
give
• Pic°(S)
8 PX )
hence
tU
dim H3(X,~)
< dim HI(S,~)
Therefore
the i n j e c t i v i t y
(*)
the e q u a l i t y
give
of
+ dim H3(S,@)
~t' ~t
in the last
together
+ d i m HI(C,~)
with relation
formula. q.e.d.
78 Consider
the m o r p h i s m s
~j
% : J(C)
#j
: AIb(S)
+ J(X)
$ Pic°(S) +
of this section.
functorial [G],
properties
maps
J(X)
~t
' ~t
defined
The W e i l m o r p h i s m
leading
to c o m m u t a t i v e
at the
satisfies diagrams
(see
n.2)
A 1 (c) ~
~A , A 2 (x)
A 2 (S)@A 1 (S )
~j J(C)
where
~A'
#
8 ,
and by
have been
introduced
the P o i n c a r e
polarization
in s e c t i o n
s u m of W e i l m o r p h i s m s .
0 the p r i n c i p a l P
on
~
A 2 (X)
~j A/b(S)$Pic°(S) ~
, J(X)
is the d i r e c t by
varieties
,
i n d u c e d by the t o p o l o g i c a l beginning
of a b e l i a n
divisors
J(X)
divisor which
AIb(S)~Pic°(S)
2.2 a n d
Moreover
on
,
wS
we d e n o t e PX r e s p e c t i v e l y
defines
Let
J(X)
a principal
m:P X ÷ J(~)
be
the i n c l u s i o n .
THEOREM
4.2.4
(i)
The m o r D h i s m s
Cj + ~jom:Alb(S)
~j0m,
• Pic°(S)
~j
induce
$ PX + J(X)
an i s o g e n y
79
(ii)
The equalities
(~jom)
@ = 40 , ~ 8
= 29
hold,
up
to a l g e b r a i c e q u i v a l e n c e .
Proof.
(i)
Look at the morphism
p = ~AOm:Px
÷ A2(X)
and the isogeny
~+p:A2(S)
stated
in section
of diagrams
(i),
~ AI(s)
~ Px÷A2(X)
2, Proposition (2) implies
A 2(S)eA I(S)eP x
~+P
(Ws'id') I
then
~ = ~j + ~jom ~
of Lemma
is also.
that one of the square
~ A 2(x)
.
Since
Moreover,
~ ~ J(X)
w , (Ws,id.) as we have
are surjective
seen in the proof
4.2.3
dim J(X)
Therefore
The commutativity
~I w
Alb(S)S P i c ° ( S ) S P x
where
2.2.4.
kero
= dim(Alb(S)
is a finite
• Pic°(S)
group,
8 PX )
so that
~
is an isogeny.
80 (ii)
Recall
the p r i n c i p a l
that
the
divisors
E@(a,b)
=
forms
@ , 8
= -%(a-b)
E8(¥,¥')
via P o i n c a r e
Riemann
(y.y')
duality.
E@
are
given
,
a,b
e HI(C,~)
,
y,y'e
Since
for all
find
(~30m)
Now,
let
Poincare
a,b
Ep
be
the R i e m a n n
P
on
,
form associated
Alb(S)~Pic°(S)
Note
and anti-linear
determined we have
= -2(a'b)
to the
Consider
the
= Ep((x,O) , (O,y))
HI(S,~)xH3(S,~)
HI(S,~)
e HI(C,~.)
form
E(x,y)
on
,
e = 40
divisor
~-bilinear
associated
by
H3(X,Z)
(~t0m(a) -~tom(b)) = -a- (b-T'b)
we
, E8
by
(see
E [MI],
and II,
that
on
H3(S,~)
for all n.9)
Ep((x,o) , (O,Y))
E
= x-y
(x,y)
is
C-linear Moreover
£ HI(S,Z)
on Ep
is
× H3(S,~)
to
81
Since
~t(x,o)'~t(O,y)
= f
( x . y ) ' i , T X = 2(x'y)
#t(x,O) -~t(O,y)
= 2 E p ( ( x , o ) , (O,Y))
we get
and
~j8 = 2P
in
NS(AIb(S)ePic°(S)) q.e.d.
To c o n c l u d e , base
surface
PROPOSITION that
q(S)
~ : P x ÷ J(X)
Proof. a E
S
4.2.5 = O
consider
the p a r t i c u l a r
is r e g u l a r
Let Then
show
i.e.
f:X ÷ S there
of principally
First we
Hi(C,VZ)
we
that
- 0 modulo
be
exists
for all
2
the
= O
a conic
bundle
such
an i s o m o r p h i s m
polarized
one has
y'~t(a)
q(S)
case when
abelian
cycles
Y
varieties.
~ H3(X,Z)
,
82
Lemma
4.2.3
gives
H3(X,Z)
and
after
Lemma
= ~t(HI(C,~))
4.2.1
ii) w e
n H3(X,~.)
can write
,
a cycle
_
y E ~t(HI(C,~))
n H3(X,~)
in the
y = ~il i ~ t ( o i - T ( a i ) )
By u s i n g
Beauville
y-~t(oi)
= 21 i , i ~ g , 2 g
y,
~t(oi)
in
formula
H3(X,Z)
then we
y-~t(a) get
~ 0 modulo
.
4.2.1 we
Therefore
Again
Putting
4,
formula
being gives
i ~ g,2g
Ua(Y)
= ~t(a)'y/2
there
exists
÷
Therefore
Poincare
cycle
E H3(X,~.)
a'-y
= u
a
duality such
(y)-y
implies that
that
i ~ g
find
21 i c Z
Beauville
- O modulo
2
i = i, ... t2g-l,
a morphism
Ua:H3(X,~)
a'
r I.l e ~,
and Lemma
~ t ( o i - T ( o i ) ) "~t(a)
Hence
form
a
,
83 for all modulo
7 E H3(X,~.)
(see
[G-HI
let
P2
has
the
now
be the
the c o m p o s i t i o n set of p o i n t s
~t(a)
isomorphisms
= 2a'
exists
2
in
,
and
PX
One
(of groups)
P2 c k e r ( ~ j 0 m ) there
T j 0 m : P x ÷ J(X)
of o r d e r
PX ~- HI(C'I~)/HI(C'~')
Thus
Thus
torsion.
Consider
Then
p.53).
since
' J(X)
~t(a)
a morphism
~- H3(X~R)/H3(X,2Z)
modulo
= 2a'
such
~:Px ÷ J(X)
torsion. that
the
diagram
~jom PX
~ J (X)
Px
commutes.
It f o l l o w s
2 ~
by T h e o r e m
4.2.4(ii) .
Hence
~ 8 = 0
achieves
the proof
torsion J(X)
free.
' PX
have
This the
same
=
(Tj0m)*e
dimension
= 40 as
in
N S ( P x)
NS(Px )
after
by T h e o r e m
is
recalling 4.2.4
that (i).
q .e .d.
COROLLARY and
let
4.2.6 ~:X'-
Let ÷ X
X + S
,
X' + S'
be a b i r a t i o n a l
map.
be c o n i c Suppose
bundles q(S)
= O.
84
Then (i)
There
abelian
exists
with
J(C i)
(ii)
Assume
Jacobians
Proof.
X
;
Then
PX
is i s o m o r p h i c r as
v a r i e t y r to a p r o d u c t
C. l
are
fundamental
locus
implies
-~ PX'
@ .HJ(C i) i
the c u r v e s of
q(S')
of
(i).
of p r i n c i p a l l y
+ J(X')
an isomorphism
consequence
s o let us p r o v e
a morphism
= 0
abelian
is an e a s y
J(X)
J(X')
to b e r u l e d .
polarized
(ii)
we have
of curves.
of
of curves.
X' = S ' x P 1
q(S)
polarized
@ ~-J(Ci) 1
Jacobians
principally
the
of p r i n c i p a l l y
varieties
P X + PX'
where
an e m b e d d i n g
According
polarized
to
[C-G]
abelian
varieties
,
which
occur
in the
~
Morover
= 0
Then
of principally
(i) b y c h o o s i n g
resolution
the h y p o t h e s i s
Proposition
polarized
of
abelian
4.2.5
gives
varieties
a n d w e are d o n e . q.e.d.
85 §5.
Some open q u e s t i o n s
The n o t a t i o n s
are as in p r e v i o u s
sections.
Let X ÷ S
be a conic bundle. (i)
Consider
the m o r p h i s m of a b e l i a n v a r i e t i e s
(~j0m) : (Px,@)
where
0, e denote
4.2.4(ii) Suppose
that
+
(J(X),e)
the p r i n c i p a l p o l a r i z a t i o n s .
(~j0m)*@= 40 in the N e r o n - S e v e r i
the map
~j0m f a c t o r i z e
We p r o v e d in group NS(Px).
t h r o u g h the m u l t i p l i c a t i o n
by
2 as follows ~j0m PX
> J(X)
PX Hence
~ is a m o r p h i s m of p r i n c i p a l l y p o l a r i z e d
a b e l i a n varieties,
so that we get a d e c o m p o s i t i o n
(*)
J(X)
= PX @ A
for some p r i n c i p a l l y AIb(S)
@ Pic°(S).
polarized
abelian variety
Then a q u e s t i o n is to c h a r a c t e r i z e
conic b u n d l e s
for w h i c h the i n t e r m e d i a t e
decomposition
in the w e a k e r
one
A isogenous
form
to
all
J a c o b i a n has a
(*) or else in the stronger
88
(**)
J(X)
Recall that
= AIb(S)
(**) holds
@ Pic°(S)
true by P r o p o s i t i o n
q(S)
= 0.
(2)
C o n s i d e r now the a l g e b r a i c
A2(X).
• PX"
By u n i v e r s a l p r o p e r t i e s
4.2.5 w h e n e v e r
representative there exists
(AX,a x) of
a morphism h x
of a b e l i a n varieties s u c h that the d i a g r a m
A2(X)
w
>
J(X)
AX
commutes,
and h x is an isogeny by 4.1.2.
p r o b l e m is to see if h x is To this p u r p o s e are i s o m o r p h i c (3)
(or if h x induces)
note that w h e n e v e r
(**) holds
an i s o m o r p h i s m . then A x and J(X)
(not n e c e s s a r i l y via hx).
A result of R o i t m a n
as:A2(S)
Again a natural
+ AIb(S)
(see [R])
says that the map
induces an i s o m o r p h i s m on the p o i n t s of
finite order
A2(S) tors. =
Alb (S) tors"
87 A q u e s t i o n a r i s i n g here is to see if an a n a l o g o u s holds
for the group A2(X),
representative
ax:A2(X)
A 2(x) tors. ~
Let us c o n s i d e r satisfy
that is if the a l g e b r a i c
÷ A x induces an i s o m o r p h i s m
(Ax) tors"
a particular
case.
the so c a l l e d A b e l - J a c o b i
that as:A2(S)
+ AIb(S)
Suppose property:
is an isogeny
the s u r f a c e
(see [M-B]).
+ A x is also an isogeny.
ker a x is a d i v i s i b l e
group
threefolds),
S
this m e a n s
follows that ax:A2(X)
all u n i r u l e d
result
(this p r o p e r t y holds
It
Since true
for
then a x is an isomorphism.
REFERENCES
[B]
Beauville, A. m~diaires,
[B-F]
:
V a r i e t ~ s de P r y m et J a c o b i e n n e s
Ann. E.N.S.
Beltrametti,
IO (1977),
M. and Francia,
P.
inter-
30!9-391. :
Threefolds
with
negative K o d a i r a d i m e n s i o n and p o s i t i v e i r r e @ u l a r i t y , N a g o y a Math.
[B-M]
Bloch,
J.
(to appear).
S. and Murre,
J.P.
types of Fano threefolds,
:
On the C h o w ~roup of certain
C o m p o s i t i o Math.
vol.
39
47-105.
[c]
Chevalley,
C.
:
A n n e a u x de C h o w et A p p l i c a t i o n s ,
S ~ m i n a i r e Chevalley,
Paris
(1958).
(1979),
88
[C-G]
Clemens,
C.H. and Griffiths,
P.A.
J a c o b i a n of the cubic t h r e e f o l d (1970),
:
The i n t e r m e d i a t e
, A n n a l s of Math.
vol. 92
281-356.
[Co]
Conforto,
F.
:
Le s u p e r f i c i e
[E]
Enriques,
F.
:
Sulle i r r a z i o n a l i t ~
dipendere
la r i s o l u z i o n e di una e q u a z i o n e a l g e b r i c a
f(x,y,z)
= 0 c~n funzioni r a z i o n a l i
Math. Ann. Bd. IL (1897),
[EC~]
Grothendieck, Alg~brique, n.28
[Fa]
Publ.
da
Zanichelli
cui
(1939).
pu~ farsi
di due parametri,
1-23.
A. and Dieudonn~,~: I.H.E.S.,
II n.8
El~ments (1961),
de G ~ o m e t r i e IV
(3 me partie)
(1966).
Farkas,
H.M.
:
generalization (1970)
IF]
razionali,
Fulton,
O ~ the S c h o t t k y r e l a t i o n and its to a r b i t r a r y
genus, Annals of Math.
vol.
57-81. W.
:
Rational equivalence
(appendix to " R i e m a n n - R o c h P. Baum, W. Fulton,
on s i n g u l a r varieties,
for S i n g u l a r V a r i e t i e s "
R. MacPherson),
Publ.
I.H.E.S.
by (1974),
148-167.
[G]
[u]
Griffiths,
P.A.
:
Some t r a s c e n d e n t a l m e t h o d s
in the
study of a l g e b r a i c cycles,
Several Complex Variables
Maryland,
185
Springer-Verlag,
Hartshorne,
R.
Springer-Verlag
:
(1970).
A l g e b r a i c Geometry,
(1977) .
GTM, vol.
52
II,
92
89 ILl
Lieberman, Geometry
D.
Oslo
Noordhoff [MI]
:
Mumford,
Intermediate
1970,
(1972),
Jacobians,
5th N o r d i c
Summer
Algebraic
School
in Math.,
125-141.
D.
:
Abelian
Varieties,
Mumford,
D.
: Prym varieties
Academic
Press
Mori,
: Threefolds
Oxford
Univ.
Press
(1970) . [M2]
[Mo]
S.
numerically [R]
Roitman, modulo
(1974),
:
rational
to A n a l y s i s ,
325-350. whose
effective,
A.A.
I, C o n t r i b u t i o n
canonical
Annals
bundles
of Math.
116
are not
(1982),
133-176.
The t o r s i o n of the group of O - c y c l e s
equivalence,
Annals
of Math.
III
(1980)
553-569. IS]
Saito,
H.
:
intermediate
Abelian
varieties
dimension,
attached
N a g o y a Math.
to cycles
J.,
vol.
75
of
(1979),
95-119. IT]
Tyurin,
A.N.
dimensional
:
The middle
varieties r
Jacobian
J. S o v i e t
of three-
Math.
13
(1980),
707-814. [B-K-L]
Bloch,
S.,
on surfaces 135-145.
Kas, A. with
and Lieberman,
D.
pg = O, C o m p o s i t i o
: Math.
Zero c y c l e s vol.
33
(1976),
MODULI
OF S U R F A C E S
OF G E N E R A L
TYPE
F. C a t a n e s e * U n i v e r s i t ~ di Pisa D i p a r t i m e n t o di M a t e m a t i c a Via B u o n a r r o t i 2, 56100/PISA
Introduction The p r e s e n t
paper
the Co n f e r e n c e , expository
follows
and
rather c l o s e l y
is t h e r e f o r e
rather
of m o d u l i
some detail mations
for surfaces
a very e l e m e n t a r y
of rational
Later on we expose light on basic these
results
ruled
and of a m o s t l y
are b a s e d
of M. F r e e d m a n ' s
survey of the h i s t o r y
and at the very b e g i n n i n g though
important
example,
results
concerning
of ours
moduli
on the theory of
(cfr.
we discuss namely
result on
[5]) w h i c h
of surfaces
"bidouble"
(~ /(2))2 ) and their deformations, recent
of the with
the defor
surfaces.
some recent
questions
covers with g r o u p
We give
problem-oriented
at
nature.
In the first part we give a very brief problem
the text of the talk given
homeomorphisms
shed
of general
covers
some
type:
(i.e. A b e l i a n
and on the a p p l i c a t i o n
of 4- m a n i f o l d s
([7]).
then a list of some problems, and in the formulaticn of one of themwe are
indebt e d
to a p r i v a t e
communication
of A. B e a u v i l l e
([I]).
While in [5] the e x a m p l e s we had c o n s i d e r e d were only b i d o u b l e covers of 1 I x ~ , we enlarge hare in the second part our c o n s i d e r a t i o n to bidouble
covers
of the r a t i o n a l
can thus explain 2.18 of covers
better
ruled
the m e a n i n g
[5]), on the o t h e r we fit t o g e t h e r
surfaces
F 2m: on the one hand we
of a c e r t a i n
exact
sequence
show how the d e f o r m a t i o n s
s m o o t h l y w h e n the base
~I
x ~I
is as follows: i For a complex space X, ~X is the sheaf of h o l o m o r p h i c
(2.7.,
of the b i d o u b l e
deforms
to F 2 m .
Our n o t a t i o n
the sheaf
of h o l o m o r p h i c
and F is a c o h e r e n t
Hi(F)
the finite
dimensional
sion,
by x(F)
sheaf of 0 - m o d u l e s we denote by .X ~ -vector space HI(X,F), by hi(F) its dimen
=di~ X ( _ 1 ) i h i ( F ) . i=0
A member
O X is
functions.
If X is compact,
*
i-forms,
of G . N . S . A . G . A .
of C.N.R..
91
For C a r t i e r divisors D,C on X, Ox(D) of the a s s o c i a t e d line bundle; a l g e b r a i c equivalence,
and
is the invertible sheaf of sections
- will denote linear e q u i v a l e n c e of divisors,
IDI will be the linear system of effective
d i v i s o r s linearly e q u i v a l e n t to D; D'C denotes the i n t e r s e c t i o n product. If X is smooth T
will denote the sheaf of h o l o m o r p h i c vector fields, X and KX, w h e n it exists, will denote a c a n o n i c a l divisor, i.e. a divisor ~x'n where n = d i m ~
such that Ox(Kx)
If X is an algebraic
(compact, smooth)
X
o
surface the g e o m e t r i c genus of
X, pg,iS h ° ( ~ 2 ) = h 2 ( O x ) , the i r r e g u l a r i t y q is h°(~xl)=h1(Ox). If M is a topological m a n i f o l d of d i m e n s i o n 4, with a given orientation, T is the signature of the q u a d r a t i c form q:H2(M,ZZ )+ZZ given by Poincar4 duality. As usual b i = d i m ~ H i ( M , ] 9 ) = dimzM i=0
is the i th Betti number and e =
(-1)lb. is the topological Euler-Poincar4 characteristic of M. 1
§ 1. Moduli of surfaces:
history and problems.
Let S be an a l g e b r a i c compact smooth surface, w h i c h we assume to be minimal
(i.e. S does not c o n t a i n curves E ~ ~I
Like the genus of a curve,
the h o l o m o r p h i c
such that E 2 = -I).
invariants K S' 2 X(O S) =
x(S)
depend only on the t o p o l o g y and the o r i e n t a t i o n of S (this last being induced by the complex structure). In fact (1.0)
~
K2 = 3 T + 2 e
[
12 X = 3 T + 3 e ,
the H i r z e b r u c h - Riemann - Roch A s s u m e that S
P
as a c o n s e q u e n c e of theorem
(cf.
[10]).
~ B is a c o n n e c t e d family of smooth surfaces,
i.e.
a) B is a c o n n e c t e d complex space -I b) p is proper and Sb= p ({b}) is smooth for each beB. It is then a classical result that all the Sb'S are d i f f e o m o r p h i c
to
each other. A c c o r d i n g to M u m f o r d Definition
1.1.
([17]) one has the following definition:
The complex space M is said to be a coarse m o d u l i space
for S if there exists a b i j e c t i o n g from the set of i s o m o r p h i s m classes {IS' ] I S' is h o m e o m o r p h i c to S by an o r i e n t a t i o n p r e s e r v i n g h o m e o m o r p h i s m }
92
to M such that for each family S (such that f(b)
= g([ Sb]))
P
~ B the induced m a p p i n g f: B
~ M
is holomorphic.
A m o d u l i space does not n e c e s s a r i l y exist,
as shows the following example,
of rational ruled surfaces. Example
1.2.
C o n s i d e r the rational ruled surfaces
n = ~ (0
18 0
1 (n)),
for n> 2. Direct computations,
which are well known,
(cf. e.g.
[12] pag.
42) give
the result that for n>2
(I .3)
h°(T]?
) = n + 5,
h1(T~
)
n In particular, m
these surfaces are not b i h o l o m o r p h i c to each other, but •
are d i f f e o m o r p h i c
iff
n
2 - v e c t o r bundles V on ~I
(1.4)
= n-1.
n
0
~ m (mod
2).
In
fact,
consider
~ V
These are c l a s s i f i e d by the and we get thus a family F
+ 0~i
(n)
(n-l) - d i m e n s i o n a l v e c t o r space B = H I ( o ~ I ( - n ) ) , -I P * B of ruled surfaces where p (b)=
~ V
is trivial
if m>n, and,
zero ~ gives a splitting of the exact sequence For b e B, let m(b) homomorphism
(m)
~ Vb:
if m=n, a non
(1.4).
by the m a x i m a l i t y of m(b)
determines a subline bundle of Vb, m o r e o v e r Roch theorem, hence V b ~ 0 --
(1.4) any
the m a x i m u m m for w h i c h there exists a non trivial
~ : O~i
Since n<2 m(b)
rank
~ 0
(Vb), V b being the v e c t o r bundle c o r r e s p o n d i n g to b e B. By (m)
1 (m(b)) 8
0~i
2 m(b) ~ n by the Riemann(n-m(b)), and ~ ( V b ) ~ ~ 2 m ( b ) - ~
< 2n, we get a d e c r e a s i n g f i l t r a t i o n B --
such
of B, for n/2<m
such that B
= {b I exists a non trivial h o m o m o r p h i s m f : O ~ 1 (m)---+ V b} m C l e a r l y Bm - Bm+l = {bl ~ (Vb) ~ F 2 m-n}' and we have seen that Bn c o n s i s t s of the origin only. proposition min
1.5.
B
m (n-l, 2(n - m)).
Proof.
and
which are e x t e n s i o n s
~ 0~i
h o m o m o r p h i s m ~:0~i
the
n
is an algebraic cone of d i m e n s i o n equal to
Bm = {bl H ° ( V b ( - m ) ) ~
By virtue of the exact sequence
0}
93
0
~ H °(v b(-m))
where
, H l (o0 ~
is g i v e n b y c u p
8(b)
B Fixing
two p o i n t s ,
(n-m))
product ={ b I
m
8(b) .... , H I (O~i (-m))
with
8(b)
0 a n d ~, in ~ 1
is n o t i n j e c t i v e }
and choosing
z on ]p1 _ {~} s u c h t h a t 0 c o r r e s p o n d s H I(0~I
(-n)) z
whereas
Since
-I
i) ii)
z
a basis
-n+1
e
H°(~ 1-
an a f f i n e
to the o r i g i n ,
{0} -{~},
for H°(O~pI (n-m))
(z-i)v z j = { zj-i 0
if
. coordinate
a basis
for
is g i v e n by the C ~ c h c o c y c l e s
....
n-1 [ b. z -i 1 i=I
b =
beB=H 1 (0~1:~1 ( - n ) ) ,
2 m < n+1,
0 ~ I ),
is g i v e n by
1,z,...
z
n-m
or if j>i or j-i_<-m
belongs or,
to
B
m
for 2 m > n+2,
if the f o l l o w i n g m a t r i x A
(b) has rank s t r i c t l y less t h a n
m
(n-m+1),
where Am(b) =
bI
b2
bn_m+ I
b2
b3
bn_m+ 2
•
•
•
bm-1
•
bm
.
•
,
•
" "
It is i m m e d i a t e n o w that B assertion
•
on its d i m e n s i o n
i
o
•
,
bn-1 is an a l g e b r a i c
m
by considering
cone,
V.
and we shall p r o v e the
= {b I the f i r s t
i columns
l
of A
(b) are l i n e a r l y d e p e n d e n t } and p r o v i n g , by i n c r e a s i n g m on i, t h a t c o d V. = m-i, the c a s e i=I b e i n g i m m e d i a t e .
induction
l
Now of
Vi-Vi_lis
covered
(m-i) h y p e r s u r f a c e s ,
by o p e n hence
sets w h e r e c o d V. i
But
if it were c o d V. < m-i, l
Pi = {bl b.=O 3 least 1, while it
for
is
V. w o u l d l
j < i - I and
easily
seen
that
it is a c o m p l e t e
intersection
< m-i. --
intersect j~ m}
the s u b s p a c e
in a locus of d i m e n s i o n
V. n p . is 1 1
just
the
origin. Q.E.D.
at
94
The preceding
example
all t h e
with n>k>0
and kHn
~ Fo
x ~I)
Fk's
the
fibre
Now
it is e a s y
cannot
is
exist
holomorphic
(~ I
for these map
The case when
feature
the
image X
a)
X
b)
the
~ X
m
m
(1977,
[8]).
m
(m-l) 2
of #
D n E6
E7
{ z 2 + x(y2
+x
=
{
y
{ Z
2
+ x
2 E8
c)
=
{ Z
+ y
+
m
n+1
=
=
for n odd. to d e f i n i t i o n
have
be constant
~m
and
surface
are
type
on a d e n s e
open
topological
set, space.
l u c k y one: variety
of the c a n o n i c a l the complete , ~ Pm-1
such
1.1.)
a non costant
is a r a t h e r
: S
in
by the
divisor
linear
system
, where
that
enjoying
R.D.P.'s
the
following
properties
(Rational
4
=
3
Double
Points)
i.e.
singularities:
= 0}
n-2) = 0 }
(y3+ x 2)
+ x
we would
to t h e h y p e r s u r f a c e
{ z
+ x 3
~ ~I
as f i b r e s
open s e t of B
5
of X = X
2
contains
(according
([2]) , w h e n m ~ 5
+ x(S),
=
Z2
+ x
and
the p r o p e r t i e s
morphism
(locally)
2
by
is a n o r m a l
m
n,m>
biholomorphic
n
K2 S
~ B
it is a q u a s i - p r o j e c t i v e
of B o m b i e r i
singularities
A
is g i v e n
P
is a H a u s s d o r f f
of g e n e r a l but
a birational
for
n
a complex
M exists
here
by the theorem
_
because
space
The key
m
space
should
of G i e s e k e r
p
surfaces,
F
and on a Zariski
for n even,
since
theorem
(1.6)
family
(mod 2),
a moduli
S is a s u r f a c e
yields
the
f: B --÷ M w h i c h
this case not only
Im KSI
that
to see t h a t
a contradiction
KS:
shows
(n>4)
0}
= 0}
5 + y
= 0}
~ = ~
is a m i n i m a l r e s o l u t i o n of s i n g u l a r i t i e s o f X, i.e. # is m b i h o l o m o r p h i c o u t s i d e t h e s i n g u l a r p o i n t s of X a n d t h e i n v e r s e i m a g e of a s i n g u l a r E 2 = -2
point
intersecting
the Dynkin
diagrams
is a
(connected)
transversally,
union whose
of c u r v e s structure
E ~ ~I
with
is d e s c r i b e d
by
95
A
D
n
n
:
0
:
/
O
O :
E6
O
.....
O
O
.....
O
r
O
O
~
O
O
o
o
o
o
o
o
o J
:
E7
o
o
~ o
:
E8
o
l ~
o
w hose v e r t i c e s
correspond
to curves,
of i n t e r s e c t i o n
of the two curves
edge.
(e.g.
The index
n in A
n
o
and w h o s e
edges c o r r e s p o n d
corresponding
) denoter
to p o i n t s
to the v e r t i c e s
the n u m b e r
of v e r t i c e s
of the
in the
diagram. The
importance
morphism
of the p l u r i - c a n o n i c a l
f: S
M appears
~ S'
as a q u o t i e n t
scheme p a r a m e t r i z i n g group
PGL
(1.7)
induces
(P). m
1.8.
ing to the
surfaces
of general
one
(cf.
Let m>5,
and set N=P
and we get an
of surfaces
previous
result.
S for w h i c h G acts
If G acts
of X
lifts
correspond faithfully
subvariety
faithfully
of M.
on S, G acts
h e n c e on its dual
representation leaving
p of G on ~ n
lifts
? of G i n d u c i n g X
m leaving
to an a u t o m o r p h i s m
an injective
to a linear
is only a finite
representations
O s ( m KS)),
of G on ~ N
representation
m desingularization),gives
N o w there
of the
group and let M G be the subset
(S)-I.
(N+1)-dimensional
any a u t o m o r p h i s m
in p a r t i c u l a r
395).
application
space H°(S,
representation
any faithful
to a finite
m
on the v e c t o r
projective
Xm and Xm,' therefore
K2,X b e l o n g
of S. Then M G is a c l o s e d
--
linearly
fixed
[3], p.
classes
as a g r o u p of a u t o m o r p h i s m s Proof.
between
has:
type with
Let G be a finite
isomorphism
in the fact that any
c l o s e d subscheme H of the H i l b e r t 2 2 ~-I of degree m K S in ~ by the p r o j e c t i v e
later use we show a n o t h e r
Theorem
lies
m
of a l o c a l l y
n u m b e r of f a m i l i e s For
X
a projectivity
In p a r t i c u l a r
Surfaces
model
homomorphism
of i s o m o r p h i s m
p, and c o r r e s p o n d i n g l y
X
m of S
(N+I)-dimensional
number
invariant.
space, a faithful
Conversely
invariant,
since
(S b e i n g a m i n i m a l
of G into A u t ( S ) ~ h e n c e representation
classes
of G.
of such projective
M G can be e x p r e s s e d
as a finite
g6
union of subsets inducing scheme
M p. Fix t h e r e f o r e
a faithful
action
of the H i l b e r t
surfaces Clearly
such a
(linear)
representation
on ~ N , and let H Q be the locally
scheme
H parametrizing
sub
of m of fixed points for the actinn of G cn H.
S, such that H p is the locus M p is the p r o j e c t i o n
m-canonical
p
closed
to the q u o t i e n t
images X
of the image of H p x PGL(N+I)
in H. To prove
that such imaqe is c l o s e d we use the v a l u a t i v e
properness family S
(cf. P
a faithful
e.g.
[9 ], t h e o r e m
~ B, and that,
By the m -th c a n o n i c a l and it suffices X* to
mapping
to p r o v e
that
that we have a l - p a r a m e t e r -1 = B -{ bo}, S* = p (B*), we have
B*
such t h a t , ~ o r
g e G,
we get a family
X
p o g = p.
f
, B,with
for every g in G its action
X ---+B x ~ N
extends
X and in such a way that g does not act as the i d e n t i t y
Xo = f-1(bo):
in fact,
X* b e i n g
hence we get a h o m o m o r p h i s m
of
4.7.) : a s s u m e
setting
action of G an S*,
criterion
dense,
such e x t e n s i o n
of G into A u t ( X o) w h i c h
,
from
on
is then unique,
is injective
by the
second property. Now,
for g e G, we get an i n v e r t i b l e
can assume
to be g i v e n by a regular
it suf f i c e s
to p r o v e
the identity. eigenvalues
that a(bo)
By continuity,
of a(t),
whose
matrix
a(t),
function
on B with a(bo) ~ 0: c l e a r l y
is i n v e r t i b l e
the e i g e n v a l u e s
ratios
therefore,
not all the e i g e n v a l u e s
eigenvalue
of a(bo),
and is not a m u l t i p l e
of
of a(b o) are limit of the
are c e r t a i n of a(bo)
for t e B*, w h i c h we
fixed
roots of unity,
are equal
and if 0 were
an
then a(b o) w o u l d be zero, a c o n t r a d i c t i o n . Q.E.D.
As we h e a r d spaces
M
g projective other
from D. M u m f o r d ' s
of curves normal
side,
of surfaces expectations general" But,
irreducible
of general
and we
have a n e g a t i v e
highly
reducible, to e x p l a i n
and let's make
Definition
variety results
type,
1.9.
much
g, the basic
not m a n y general
in order
notation
of genus
lecture,
answer:
with
is k n o w n
fact b e i n g
of d i m e n s i o n are k n o w n
e.g.
Let S be a surface
spaces
that too o p t i m i s t i c
a lot of c o m p o n e n t s
same h i s t o r i c a l
is quasig (g~2) ; on the
the m o d u l i
these m o d u l i
all this more precisely,
the m o d u l i
that M
3g-3
about
shall show here
about
spaces
are
of d i f f e r e n t let's
"in dimension.
introduce
some
remark.
of general
type:
then
the n u m b e r
of
97
moduli of S, d e n o t e d by M(S), at the point
is the d i m e n s i o n of the moduli
space M
[S] c o r r e s p o n d i n g to S.
M. N o e t h e r ([18])in 1888, under very special hypotheses,
p o s t u l a t e d for
M a formula which in our t e r m i n o l o g y reads out as M = 10 X This formula is v e r i f i e d quite seldom
2 K 2.
(especially since M is a p o s i t i v e
integer, whereas the right side can be very negative,
even for complete
intersections), but it is the m e r i t of F. Enriques to u n d e r s t a n d that 10 X -
2 K 2 should give a lower b o u n d for M in the case of non ruled
surfaces. In fact Enriques gave two proofs
(see e.g. his book [6] ,p.204-215, especia !
ly the historical note on page 213) w h i c h were b o t h incomplete, fact relying on some a s s u m p t i o n s w h i c h did not hold true. proof Enriques assumed to have a surface F c 3
and in
In the first
with o r d i n a r y s i n g u l a
rities, of degree n, and with double curve C: he a s s u m e d that the chara cteristic
system
(cut on the n o r m a l i z a t i o n of F by a d j o i n t surfaces of
degree n) should be complete, and this is not true in general as was shown by K o d a i r a
in 1965
([11]); similarly in the second proof it was a s s u m e d
that the c h a r a c t e r i s t i c should be complete,
system of plane curves w i t h cusps
and nodes
an a s s e r t i o n w h i c h was d i s p r o v e n by Wahl in 1974
([22]), relying on the e x a m p l e s of Kodaira more thorough discussion,
(we defer the reader,
for a
to the a p p e n d i x to C h a p t e r V of Zariski's book
[ 25], w r i t t e n by D. Mumford). A proof finally came in 1963, through the t h e o r e m of K u r a n i s h i
([13])
c u l m i n a t i n g the theory of d e f o r m a t i o n s of complex structures due Kodaira and Spencer.
to
! I
Let
X
P, B be a c o n n e c t e d family of smooth m a n i f o l d s and boeB: -I the fibres X b = p ({b}) are said to be d e f o r m a t i o n s of Xo=Xbo.
then
Any h o l o m o r p h i c map f of a complex space T into B, with f ( t o ) = b o , i n d u c e s another family of d e f o r m a t i o n s of Xo, namely the fibre product T XBX. A family of d e f o r m a t i o n s
(X,Xo)
if, for every other d e f o r m a t i o n
P
,(B,b o) is said to be s e m i - u n i v e r s a l
(Y,X o)
g
,(T,t o) the r e s t r i c t i o n to a
s u f f i c i e n t l y small n e i g h b o u r h o o d of t o is induced by a h o l o m o r p h i c map f : T->B whose d i f f e r e n t i a l at t o is u n i q u e l y determined; be universal
it is said to
if m o r e o v e r such a f is always unique.
The theorem of K u r a n i s h i asserts that a s e m i u n i v e r s a l d e f o r m a t i o n exists
98 (it is then unique by its defining property), base B is a germ of analytic
subset of
and moreover
(HI(Xo
that its
),0) defined by h2(Xo
,TXo TXo) equations vanishing of order at least two at the origin. Later Wavrik universal,
([23]) proved that,
what implies
the germ of M at
if H°(TXo)=0 , then the deformation
[Xo~ is biholomorphic
to the quotient B/Aut(X o)
e.g. in the case of Galois covers whose deformations the action of Aut(X o) on HI(TXo)
type, Aut(S)
of pluricanonical
being the Lie algebra of a finite group. solution to Enriques' theorem:
(though,
are all Galois covers,
need not be effective).
Now, when S is a surface of general is another application
is
that if a moduli space M exists for Xo, then
is a finite group
embeddings), Deformation
inequality via the Hirzebruch
if a surface X is not ruled,
hence H°(Ts)=0, theory gives a
- Riemann
then H°(Tx)=0,
(this
- Roch
and M = dim B, if
M exists. Clearly one has, by the previous (1.10) but,
remarks on B,
h I (Tx) - h2(Tx ) ~ dim B = M ~ h I (Tx)
since h°(Tx)=0 , the left hand side is -X(Tx),
Hirzebruch
R.-R.
One drawback of
theorem, (1.10)
only on topological
hence
(1.10)
i.e.
10 X - 2
is exactly Enriques'
K 2 by the
inequality.
is that the upper bound for M does not depend
invariants:
hovever,
since
bY Serre duality
h2(Tx ) = hO(~ ~ ~ ~2 X )' the right hand side is 10x-2K 2 + h ° ( ~ an upper bound on h ° ( ~
~ ~),
® ~2X ) ' so it is enough to give
and in the case e.g. of surfaces of general
type this can be done via exact sequences
restricting
the sheaf ~ I ~ 2
'
to a smooth curve in One gets
IKI or
S
S
ImKl.
(theorems B and C of [5]) the upper bounds
(1.11)
M <
10 X + 3 K 2 +
(1.12)
M <
I0 X + q + 1
These extimates
108
(in general)
if S contains a smooth canonical
appear to be too crude and an interesting
roughly
speaking:
(1.13)
what is a s y m p t o t i c a l l y
the best upper bound for M?
question
curve C. is,
99
I will return later to better bounds for irregular surfaces, m o m e n t let me remark that, vo's t h e o r e m , N o e t h e r ' s M i y a o k a - YaH,
for the
for a surface of general type S ,by C a s t e l n u o -
inequality,
the topological
and the i n e q u a l i t y of B o g o m o l o v -
invariants K2,X
are subject to the fol-
lowing inequalities : (1.14)
~} 2 _> I ,X _> I K2 > 2 X - 6
2 ! 9 x. It is possible therefore that, as K2,X
~ + ~
one may have d i f f e r e n t
best upper bounds a c c o r d i n g to the limiting value of the ratio K2/X between 2 and 9. One may ask however w h e t h e r the m o d u l i space is p u r e - d i m e n s i o n a l :
we
proved r e c e n t l y that this is not true, and that M can attain a r b i t r a r i l y many d i f f e r e n t values for o r i e n t e d l y h o m e o m o r p h i c
surfaces.
More precisely, we p r o v e d
([5] theorem A)
(1.15)
integer n there exist integers 0<MI<M2<
for each positive and h o m e o m o r p h i c
simply-connected
... M n
surfaces of general type
S I, ... Sn, such that M(S )=M..l i An i m p o r t a n t remark is that the surfaces we c o n s i d e r are such that the c a n o n i c a l map is a b i r e g u l a r embedding,
and their invariants K2,X are
quite "spread" in the region defined by
(1.14), so that these e x a m p l e s
should be c o n s i d e r e d the rule rather than the exception. Let me sketch b r i e f l y the idea of the proof, w h i c h consists of 3 basic ingredients. Step I:
If Sl, S 2 are simplyconnected, they are o r i e n t e d l y h o m e o m o r p h i c
Step II:
a)
K
b)
K
S. 1 S. l
e 2 P
have equal K2,X, and K 2 # 9, if and only if either
(S.) l
(i=I,2)
{ 2 P. (S.) ic l
(i=I,2)
ic
Find families of surfaces, with the p r o p e r t i e s stated in step I, d e p e n d i n g on many integral parameters, of those
K2,x,M.
and compute in terms
Step III:
Show that one can fix
K2,X and obtain d i f f e r e n t v a l u e s
MI,... M
for M: this is a number theoretic problem, solved n by E. B o m b i e r i (cf. the a p p e n d i x to [5]), so that I will not talk about this in a C o n f e r e n c e on A l g e b r a i c Geometry.
Step I was suggested by B. M o i s h e z o n recent deep t h e o r e m of M. F r e e d m a n
(1.16)
and depends almost e n t i r e l y on the
(cf.
[7]).
If Sl, S 2 are s i m p l y - c o n n e c t e d compact o r i e n t e d d i f f e r e n t i a b l e 4-manifolds with the same i n t e r s e c t i o n form on H 2 ( S . , ~ ), then l they are (orientedly) homeomorphic.
and on the theorem of Yau (1.17)
([24])
if K 2= 9 X and K is ample,then the universal cover of S is the unit ball in ~ 2
In fact, by
(1.0), K2,X d e t e r m i n e the rank and the signature of the
u n i m o d u l a r q u a d r a t i c form q : H 2 ( S , ~ ) un~larinteqral and p a r i t y
quadratic
~
, and it is k n o w n that indefinite
forms are c l a s s i f i e d only by the rank,signature
(q being even iff for each x, q(x)~0
(mod 2), being odd other-
wise). T h e r e f o r e F r e e d m a n ' s result applies p r o v i d e d that q is not n e g a t i v e or p o s i t i v e definite. But q is n e g a t i v e definite o n l y for surfaces of class VII w h i c h have bi=I, while it is p o s i t i v e d e f i n i t e c o r o l l a r y of Yau's t h e o r e m
(1.17)
if and only if S = ~ 2 , as an easy
(cf.
[14],
[21]).
Step II c o n s i s t s in c o n s i d e r i n g b i d o u b l e covers of ~I
x ~I
and study-
ing their small deformations: we will return to this point in the second paragraph, where we shall consider,
using the results of
[5], the more
general case of b i d o u b l e covers of F 2 n . Anyhow
1.15 shows in p a r t i c u l a r that, fixing
of S, and varying the complex structure
the h o m e o m o r p h i s m type
(which n e c e s s a r i l y gives a
surface of general type if K 2 > 10), the number of moduli M varies
in
an interval whose size grows to infinity with K2,X . (1.18)
How many irreducible c o m p o n e n t s does M K 2 , x , ( t h e union of the m o d u l i spaces of surfaces with K2,X f i x e d ) , h a v e at most?
101
(1.19)
Is it true also that the number of c o n n e c t e d c o m p o n e n t s of M is 2 unbounded, as K ,X - - ÷ ~?
We remark here that c o n n e c t e d c o m p o n e n t s of M c o r r e s p o n d to c o n n e c t e d c o m p o n e n t s of the subscheme H of the H i l b e r t scheme m e n t i o n e d before (1.7), hence two surfaces S,S' are
such that their classes IS]
, IS' ]
b e l o n g to the same c o n n e c t e d c o m p o n e n t of M if and cnly if they are deformation of each other:
in p a r t i c u l a r they m u s t be diffeomorphic.
One could have therefore made a d i f f e r e n t choice for the "moduli space", c o n s i d e r i n g only Mdiff,
i.e. the union of the c o n n e c t e d c o m p o n e n t s of
M c o r r e s p o n d i n g to d i f f e o m o r p h i c
surfaces, and a s k , r e g a r d i n g M d i f f , s i m i l a r
q u e s t i o n s to those p o s e d for M, i.e. p u r e - d i m e n s i o n a l i t y ,
etc..
As a m a t t e r of fact, though b i d o u b l e covers can be d e f o r m e d until the branch
locus is a u n i o n of lines
singularities),
(one gets then
surfaces with only A l-
even in this last case it is not easy to tell d i r e c t l y
~lether h o m e o m o r p h i c
surfaces are d i f f e o m o r p h i c :
we have not
pursued
this, also there is some hope that F r e e d m a n ' s result can be m ~ d e as to imply that the two given 4-manifolds
stronger
should be diffeomorphic°
Let's go back now to the last piece of history:
in 1949 G. C a s t e l n u o v o
([4 ]) c l a i m e d that (1.20)
For an i r r e g u l a r surface S w i t h o u t irrational pencils,
the number
M of m o d u l i is ~ pg + 2q. To explain what this means, we recall the c l a s s i c a l t h e o r e m of Casteln u o v o - De (1.21)
Franchis
A s s u m e that n1,~ 2 are i n d e p e n d e n t sections of H ° ( ~ ) s u c h
that
~I ^ ~2~0: then there exists a m o r p h i s m f:S ÷ B, where B is a smooth curve p o s s e s s i n g two l-forms
~i,~2 e H°(~B )such that
H i = f*(~i ) . Now,
such a map f: S ÷ B is called a pencil, whose genus is, bydefiniticn,
the genus of B, and an irrational pencil is just a pencil of genus at least one. So, if an irregular surface does not have irrational pencils, all its image under the A l b a n e s e map surface,
(hence q~2!).
first of
~: S + A = H° (~I"V'HsJ / i ( S , ~ ) is a
102
Conversely, and
is not
it is e a s y
to see that
the J a c o b i a n
of a c u r v e
Unfortunately
Castelnuovo's
ing c o u n t e r e x a m p l e s we w a n t some
where
to s h o w h e r e
sense
then
claim
S does
is false,
variety
not have
as we
showed
in
keeping
q
[5],
D)
it is p o s s i b l e
th.
in v i e w
that
fixed,
like
of the C a s t e l n u o v o
A
is s i m p l e
irrational
M grows,
(cf.
his a s s e r t i o n ,
if the A l b a n e s e
[5]
4 pg;
pencils. , exhibit-
anyhow
to r e s c u e
in
- De F r a n c h i s
theorem.
Theorem
1.22.
If q~3 a n d
(n I ^ q2 ) is a r e d u c e d this
last
into
an
equality Abelian
Proof.
with
0
supp
a sequence
(1.24)
0 the
such
t h e n M ~ pg
that
C = div
+ 3q - 3 a n d K 2 ~ 6 X ,
if the A l b a n e s e
I
~ OC support
since
of A has
F
~
h°(~)
~ F
By the m u l t i p l i c a t i v i t y sequences,
e H°(~I)
map
is u n r a m i f i e d
sequence
ms
~
= C, and,
F, h e n c e
where
curve,
if a n d o n l y
an e x a c t
S
(F)
~i,~2
3-fold.
~ 02
"
exist
irreducible
holding
n1,~ 2 d e f i n e
(1.23)
there
~ 3, we h a v e
, A
a non
zero
of
section
, 0
dimension
of g l o b a l
>0
zero.
Chern
classes
with
respect
to e x a c t
we o b t a i n
c2(A) H e n c e c21 -> c2'
i.e.
= - length(A)
K 2 _> 6 X, and
2 (c I - c2).
= -
if e q u a l i t y
holds
A=0,
F~O c ~ q = 3
and
~I is g e n e r a t e d by g l o b a l s e c t i o n s . The a s s e r t i o n a b o u t M f o l l o w s by S t e n s o r i n g (1.23) and (1.24) w i t h ~2 b o u n d i n g h ° of the m i d d l e t e r m w i t h S' the sum of the h°'s of the two o t h e r terms, a n d h ° ( O c ( K ) ) w i t h
pg+q-1.
Q.E.D. Remark
1.25.
by g l o b a l
The
hypotheses
sections
Castelnuovo's
error
([19]):
e.g.
Severi
pencils
of g e n u s
outside
of
1.22 are v e r i f i e d
a finite
that
q the s e c t i o n s
if ~I is g e n e r a t e d S
set of points.
in f a c t w a s b a s e d claimed
e.g.
on
some w r o n g
for a s u r f a c e of H°(~!)
would
results
S without have
of S e v e r i
irrational
no c o m m o n
zeros,
103
what is not true
(see
[5] for a discussion
In the same paper Severi deduced following
statement,
and counterexamples).
from these incorrect assertions
the
whose v a l i d i t y we have not checked and we pose then
as a problem (1 .26)
Is it true that for an irregular
surface w i t h o u t
(minimal)
ir-
rational pencils K 2 > 4X? (I .27)
Also,
it is an interesting
hypotheses
question
of 1.22, C a s t e l n u o v o ' s
for us whether,
under the
inequality M ~ pg+2q holds:
looking at the proof we see that it would be indeed the case if h°(Oc(K))
could be bounded by pg+2.
has irrational pencils,
and this inequality
question posed by Enriques (1.28)
whenis the dimension
This is not true if S is related to a
([6] page 354):
of the paracanonical
system
{K} less than
or equal to pg? We recall that the paracanonical consider
the subscheme
in S algebraically the irreducible system
system can be defined as follows:
[K] of the Hilbert
equivalent
component
scheme consisting
to a canonical
of curves
divisor K, and consider
{K} of [K] which contains
the complete
linear
IKI.
At the conference we posed the problem whether irrational pencils"
would
more,
under those assumptions,
i.e. whether,
should be HI(s,n)=0,
imply
"S without
dim{K} ~ pg, and ideed we asked also for D E P i c ° (S)-{0}
it
a fact which implies dim[K] ~ pg.
This latter has been answered negatively an example where
the hypothesis
by A. Beauville
([I]) who gave
[K] has dimension bigger than pg. His example
is as
follows: (1.29)
Let B, A, be Abelian varieties
of respective
dimensions
g and q,
an element of A-{0} with 2~=0, and let i be the fixed point free involution
on B x A such that i(b,a)
Let X be the quotient m a n i f o l d B x A/i: as 0 x • 0X(n), where It is easily
the direct
= (-b, a+~). image of 0Bx A splits
2qH0 but q is not a trivial divisor.
seen that hl(0x ) = q, h1(0X(q))=g,
and that A/~ is the
104
A l b a n e s e variety of X. Taking an embedding of X by a s u f f i c i e n t l y very ample linear system, and intersecting X w i t h a general linear subspace of c o d i m e n s i o n
(g+q-2),
one gets a surface S whose A l b a n e s e v a r i e t y is just A/e, and w i t h h I (0S (~)) =g. But then,
if g>q, the d i m e n s i o n of the linear system
IK S +nl is pg+(g-q),
> Pg. Clearly, as we r e m a r k e d before, it is simple,
if A is not isogenous to a J a c o b i a n and
S has no irrational pencils.
In this example,
the system
IK] consists of
IK + ql and {K}
, which has
d i m e n s i o n pg, in fact H I ( 0 s ( e ) ) = 0 for e ePic, s), E~ 0,n, since on an A b e l i a n v a r i e t y Y the only divisor ~ in Pic°(Y) w i t h H1(0y(6)) £ 0 (cf.
~ 0 is
[16]).
To end with this first part, let m e m e n t i o n t w o m o r e p r o b l e m s w h o s e solution I'd like to see.
(I .30)
It is known
(cf. e.g.
t20 ], page 402 and foll.)
that, given any
finite group G, one can find, for each n>2, a variety X of d i m e n s i o n n w i t h ~I(X)=G. proved
(]5], Cor.
In the case where G is abelian I have
1.9) the stronger statement that for any simply-
c o n n e c t e d v a r i e t y Y of d i m e n s i o n n>2,
there exists an abelian
i
cover of Y w i t h group G n such that nI(X)=G.
I guess that someth
ing similar could be done for any finite group G, so that, particular,
in
"every finite group is the fundamental g r o u p of
infinitely m a n y surfaces". This last q u e s t i o n is a r e c u r r e n t one when one wants to d e s c r i b e explicitly some p a r t i c u l a r classes of surfaces. We recall that the p l u r i c a n o n i c a l model X of a surface S of general type is isomorphic to S if and only if the c a n o n i c a l bundle of X is ample,
i.e.
if and only if there are no curves E ~ ]i)I with K-E=0
(<=>E2---2)
(these are the curves coming from the r e s o l u t i o n of R.D.P's). It is not clear to me
wh~ther these curves can be stable by deformation,
i.e.. (1.31) DO there exist irreducible c o m p o n e n t s
Z of some m o d u l i space of
105
surfaces bundle
R. K l o t z result
has
of g e n e r a l
K S is n o t
announced
in p a r t i c u l a r
F of a u t o m o r p h i s m s only
R.D.P.
as
of M o s t o w
2.1.
group
Let
covers
/2) 2
smooth
7: S
t~t
Let
ball
(these
of r a t i o n a l
cover
A bidouble
~3 be the
= S/ai,
and
3 non
let
[S] e Z the c a n o n i c a l
if K is n o t
ample:
subgroups
F are
rigid,
by the
finite
cover
theorem
w: S
surfaces.
, X is a G a l o i s
cover
is said
to be
smooth
with
if, m o r e o v e r ,
S
bidouble trivial
1
~
cover
where
involutions X be
the
S,X,
are
surfaces, and 2 g r o u p (2Z /2)
in the induced
double
cover•
l
locus
Fix(o.) of f i x e d p o i n t s for ~. c o n s i s t s of a s m o o t h d i v i s o r 1 1 and a f i n i t e set N~ : it is c l e a r t h a t R= R I + R 2 + R 3 is the r a m i f i
l
1
cation
~(R.) = D. is a s m o o t h d i v i s o r , l 1 is the b r a n c h locus of 7. 7. Xi ~ X is b r a n c h e d on D. + D k ({ i,j,k} = {1,2,3} , h e r e 3
divisor
of
7, that
following)
; therefore
since
(nodes),
corresponding
to
12.2)
In
this
are no d i s c r e t e c o c o m p a c t s u b g r o u p s 2 D in ~ w i t h D/F not s m o o t h a n d w i t h
ruled
7. : X.
l
R.,
that K2<9X
there
the u n i t
, X be a s m o o t h
02,
The
for e a c h
varieties.
let oi, X.
the r e s u l t
singularities
A bidouble
G=(~
X, are
that
[15]).
§ 2. B i d o u b l e
Def.
such
ample?
says
of
type
the d i v i s o r
the the
D has n o r m a l
only
points
divisors
cover
of X in 0x(L'I ) b r a n c h e d
(2.3)
that
.1.
2Li
D=DI+D2+D3
and
in the
o f Xi a r e
Al-points
we h a v e : -1
N' = ~ (D n D k)_ , a n d t h e r e i j -= D-+Dk'3 so t h a t X.z is the d o u b l e
on D j + D k.
3 7, 0 S ~ 0 X ~( • 0 x ( - L i ) ) , a n d i=I
that
on X
D k + L k -- L i + L.. 3
To d e s c r i b e following
L.1 on X s.t.
then
i n N~,
crossings,
exist
[5 ] it is p r o v e n
singularities
and
more
explicitly
notation:
the a l g e b r a
x.1 is a s e c t i o n
structure
of 0 x ( D ~)
such
of 7, 0 S we use that
div(xi)=Di,
the zi
106
is a s e c t i o n
of 0s(R i) w i t h
div
(zo)=R.. i l s e t t i n g w.l = z.3 Zk, w i is a s e c t i o n
2 Then
(zj Zk)
= xj Xk,
and,
of
0 S(~* L i) w i t h 2 W.1
and w.
is p r e c i s e l y
x j xk
=
the square
root e x t r a c t e d
through
the c o v e r
~..
1
1
Conversely, bidouble
in the rank-3
bundle
V = ~ 0 (L.) one can c o n s i d e r X l i=I by the e q u a t i o n s
cover d e s c r i b e d
12.4)
w
= x
i
the
xk
j
x k w k = W i Wj and
([5], prop. Di,
divisors Def.
2.5.
exist
2.3)
Li,
all smooth b i d o u b l e
satisfying
A surface
sections
is d e f i n e d
(2.2),
S' is c a l l e d
Yi of 0 x
(Di-Li),
in V by the f o l l o w i n g
(2.6)
covers
arise
in this Way
from
(2.3). a natural
deformation
x'. of 0 x ( D j) 3 equations
lw2i : ,yjwj +
of S if there
(i,j=I,2,3)
such that S'
'k2wk ÷
j Wk - xi wi + Yi wi" Since n a t u r a l important
deformations
to k n o w
is g i v e n by the f o l l o w i n g Theorem
0
2.7.
There
÷ H ° (Ts)
are p a r a m e t r i z e d
to w h i c h
exists
result
Remark
2.6.
In
Spencer
[5] it is
is such and the Di's m o v e We are g o i n g
to a p p l y
2.19 of
rise:
it is
the answer
[5]).
sequence
3 , @ H°(0D. (D i) S i=1 1
H ° ( ~ * T x)
Im ~ = K o d a i r a
(thm.
an e x a c t
÷ H 1 (T S)
and
by a smooth v a r i e t y ,
subspace of H 1 (T S) they give
0D. (D i - L i ) ( l
)
, H 1 (z* T x)
image
of the n a t u r a l
also p r o v e d in a pencil
(2.5),
that with
deformations.
S is s i m p l y transversal
connected
if X
base points.
in the case w h e n X = r 2m. We c o n s i d e r
then
107
the f a m i l y (1.4)
the
F
P
+ B obtained
trivial
line
divisor
S c
F which
For each
b in B , S I F
from
subbundle
(1.4)
for n=2m.
of V b d e t e r m i n e s
is a s e c t i o n
In the e x a c t
a relative
of the p r o j e c t i o n
of
b = S b i S a section of F b
with
F onto
a relative
S 2 = 2m. b The p r o j e c t i o n g of
Cartier
F onto
normal
sequence
B x
bundle
.
~0
1(2m),
hence
divisor group
with
basis
given
section
of
~I
- k)
= Sb -
Lemma
a2>0
+ a2,
(m - k)Y b)
+ a 3 Yb'
if al,
Proof:
! ISbl
that
~ ~I)
(2.8)
with
is a s m o o t h
then
the
then
of f i b r e s
restrict-
not
a3 ~
a fixed 0. W e
section
system
Db-Y b =
a1((2m) part
of
-
of
Fb
(m+k))+a2=
IDbl
then
(S~) 2 = 2k.
F b ~ F2k
IDbl
either
set for c o n v e n i e n c e
with
on
smooth
of the r u l i n g
(m + k ) Y b ) =
divisor
linear
has
(0
no b a s e
and
points
if
a 3 ~ 0.
has no b a s e p o i n t s
0
Moreover,
(m + k ) Y b ) i s
+ a3 Yb'
this
a I, a 2 e ~ : w h e n
+ k ) Y b is an e f f e c t i v e
Db(S b -
If D b is an e f f e c t i v e
D b H a I S~
S~
Sb-(m
a n d D is a u n i o n
(S b -
(m - k)Yb:
2.7.
and o n l y
if
to p ) C a r t i e r
is a free a b e l i a n
D b is e f f e c t i v e ,
since
(w.r.
Yb-Sb=1.
+ a 2 Y, w i t h
since
divisor
. If a I > 0, also,
D b H a I (S bS~
Yb2 = 0 ' and Pic(F)
P H aiS
F b , if the
induces
Y, S: m o r e o v e r
' k -<m,
F2k
= a I > 0, or ai=0,
= a1(m
by
now a divisor
it to F b
onto
also
V= g*(0~1(1)).Clearly
Consider ing
~I
~ H°(OFb clearly
)
by the
, H°(OFb
IYbl
has
no b a s e
following
(S~))
exact
~ H°(Oml
sequence
(2 k))
(notice
~ 0.
points. Q.E.D.
In the p r e v i o u s H ° (Fb, k(b)
OF
discussion
(Db))
= m(b~-m
~
(cf.
we h a v e
I if a n d o n l y I .4. and
foll.
also
seen
that,
given
P ~ a I S +a2V,
if a I ~ 0, a 2 ~ -a I (m+k , O
< m
(and k ( b ) = m
(b)), only
where for
b=0). Let =
K be the r e l a t i v e
(K b
(2.9)
+
S b)
Sb
=
canonical
divisor
-2,
K - - 2 S
+
(2m - 2)
Y.
of p:F
~ B:
since
Kb-Y b =
108
By Serre d u a l i t y Moreover,
then H 2 ( O F b
(Db))=0
if D b is e f f e c t i v e , O
it f o l l o w s
÷ OFb
by the e x a c t
(-Db)
~ OFb
sequence ~ ODb
~ O,
that H I ( o r
or a I > 0, a 2 ~ - a 1 ( m - k ( b ) ) by Serre d u a l i t y
= HI(Orb
HI(OFb
Corollary
2.10.
a 2 _> -2.
Then
Let P
(-D b - K b ) ) ) =
and is t h e r e f o r e
be the d i v i s o r
Rip,(OF(P))=0
= 0 if a I ~ 0,
for i=1,2,
aiS
+ a2Y
p,(OF(P))
, and assume is l o c a l l y
a I >0,
free of rank
ma I (a I + I) + (a I + I)(a 2 + I). By the R i e m a n n
hi(O]~ b (Db))=0
= x(OFb
(Db))= H I ( O F b
+ ~ 2(m-k(b)).
to
Proof:
2 k(b).
(-((ai+2)S b + (a2+2-2m)Yb)
a 2 + 2 + a1(m-k(b))
equal
divisor.
(-Db))=0 if IDbl has no base p o i n t s or it has a b and c o n n e c t e d g e n e r a l m e m b e r (i.e., in v i e w of 2.7, ai=0, a2=I
reduced
Again
if D b is an e f f e c t i v e
- Roch theorem
for i=1,2,
hence
for i=0 one o b t a i n s
+ (a 2 +2-2m) V) = I+ ~I [ a I (a I + 2 ) - 2 m +
(a 1+I) (a2+1)
The r e s u l t
follows
chap.
12.11,
IIT,
then page
+ a2(a1+2)
considerations,
h° =
1 I = I + ~ (Db-(Db-Kb)) = I+ ~ (aiS + a2V)
(Db))
= m a I (a I+I)
and the p r e v i o u s
((ai+2)S
+
+ a I (a2+2-2m)] =
.
from the Base c h a n g e
theorems
(cf. e.g.
[9],
290). Q.E.D.
L e t g: X = F bundle T
v
~ p I be the c a n o n i c a l p r o j e c t i o n : then the t a n g e n t n T x can be w r i t t e n as an e x t e n s i o n of two line b u n d l e s , w h e r e
is the s u b b u n d l e
(2.11)
O
of vectors t a n g e n t ~ T
v
, T
to the fibres , g*
X
In the case of X = ~ b' an easy c o m p u t a t i o n Tv ~ O F
(2 S b- 2 m Yb ) , hence, b
if
(T~I)
, 0.
gives
L --- d I S+
of g
d 2 Y,
109
then hi(Tv(-Lb)) = h i ( O F
((2-dl)S b -(d 2 +2m) Yb )) b
is 0, for i=0,I,
as soon as d I ~ 3, d 2 ~ -2m.
As a consequence
we obtain:
(2.12)
Hi(T x (-Lb))=0
Proposition universal Proof.
2.13•
if d I ~ 3,
The family
deformation
It suffices
of F
F
d 2 ~ 0, i=0,I. P
, B induces
the germ of the semi-
n
to show that the Kodaira
- Spencer
map p: TB, 0
,
H I ( F o , T F ) is an isomorphism. Let V be the vector bundle on I ° B x ~ such that F = • (V) (cf. (1.4)). The relative tangent bundle of p, TFI B fits into an exact O
+ Tu
in concrete
and then on
choose
F I ~ I_{~}
bn_1,
(y',Yl,bl,... v+
hence
~
O
an affine
coordinate
z on
- {~},
we have coordinates
!
coordinates
i )
I
on
terms,
(Y0' YI' bl . . . . .
" ~ g* (T
T F IB
g being the projection Now,
sequence
v
Yo = Yo Yl
z),
whereas !
bn_1,z
),
on F I ~ I_{0 } we have
with
z' = I/z I
n-1 n-i E b.1 z i=I
Y" = Yo + Yl !
n-1 -i Z b.z i=I z
-n
Yl = Yl .z Since
3
P(-J-C--~ ) is the difference
of the two liftings
of
according bi
to the two given coordinate (2 14)
patches,
P( ---~ ) = (Y4 " zn-i)
•
~ b,
~
we obtain -
~Yo
z-l)' (Yl
~yo
1
These are
(n-l) elements
in H I (U, T U ),U being the cover given by the
two open sets above• An easy computation HllFo
shows that,
for b=0,
these elements
are a basis
, T F o )Q.E.D.
of
110
Let now X be a smooth b i d o u b l e cover of F = F 2 m
c o r r e s p o n d i n g to the
divisors L1,L2,L 3 and b r a n c h e d on the divisors DI, D2,D 3. If L i is H a.l S + b.Y,1 we shall say that X is of type
(al,b I)
(a2,b2),
(a3,b 3 ) • Now,
if a
> 3 i
--
is ~ H I ( T ) Moreover,
b. > 0 '
1
for each i
--
by
(2.12) Hl(n * T
) ~ H1(w,w * T F )
'
.
consider then the family
on it, the divisor
F
P ~B of d e f o r m a t i o n s of F , and,
ii ~ ai S + bi Y' Di ~ ~(I (aj+ak)S + (bj+bk)Y)
: then
the direct images of the a s s o c i a t e d invertible sheaves in F are locally free
(and
R i p, = 0 for i ~ 1,2).
On the other hand p, OF(? i -i )l is locally free if
(2.15.i)
~ a j + a k - 2a i ~ 0 bj+b k
(2.15.ii)
2 b i ~ 2,
but also
(it is then equalto zero)if
aj + a k - 2a i < 0.
If, for each i=I,2,3,
either
(2.15.i) or
(2.15.ii) holds,
choose a t r i v i a l i z a t i o n of p, OF(Di) , p, OF(Di-ii) Then one has a v e c t o r space
then one can
on B.
U and, for each b E B, u £ U, sections Yi of
OF(Di-ii),
x~ of O ( D ) : a c c o r d i n g to (2.6) one defines a family of ] fF j deformations X ;Bx U which, r e s t r i c t e d to {O} x U, gives the natural d e f o r m a t i o n s of X. In v i e w of theorem 2.7 and of p r o p o s i t i o n
2.13, the
a s s o c i a t e d Kodaira - Spencer map is surjective. Thus we get the following. Theorem
(2.16)
(ai,b i)
(i=I,2,3) with
moduli
Let X be a smooth b i d o u b l e cover of F (2.15)
= F2m
of type
i) or ii) holding for each i. Then the
space of X c o n t a i n s only one
(unirational)
irreducible c o m p o n e n t
passing through X, and its dimension equals Proof.
~ h ° ( O ~ (D.))+h°(O F (D.-L.))-6. 1 1 1 i f In v i e w of the p r e c e d i n g d i s c u s s i o n the family X ~ B x U,
which is induced by a m o r p h i s m h of B x U universal deformation,is
such that h is of
, HI(Tx)
from the semi-
maximal rank at the origin of
the v e c t o r space B x U. Therefore the s e m i - u n i v e r s a l d e f o r m a t i o n has as basis an open n e i g h b o u r h o o d of the origin in HI(Tx) , m o r e o v e r then
111
B x U d o m i n a t e s an affine n e i g h b o u r h o o d of
IX] in its m o d u l i space
M.
The a s s e r t i o n r e g a r d i n g the d i m e n s i o n follows from theorem 2.7, since, by
(2.12) , h I (~* T
)- h °(~*T F ) = h I (T F ) - h °(TI~ ) = - 6. Q.E.D.
References
[I] Beauville, A.
: Letter to the Author of September
1982.
[2] Bombieri, E. Publ. Scient.
: "Canonical Models of Surfaces of General Type", I.H.E.S. 42(1973), 171-219.
[3] Bombieri, E. - Husemoller, D. : " C l a s s i f i c a t i o n and E m b e d d i n g s Surfaces", Alg. Geom. A r c a t a 1974, Proc. Symp. Pure Math. 23, A.M.S. P r o v i d e n c e R.I. (1975), 329-420. [4] Castelnuovo, G. : "Sul N u m e r o dei Moduli di una S u p e r f i c i e Irregolare" I, II, Rend. Acc. Lincei VII (1949), 3-7, 8-11. [5] Catanese, F. : "On the Moduli Spaces of Surfaces of General Type", to appear in Jour. of Diff. Geom. [6] Enriques, F.
: "Le Superficie A l g e b r i c h e " ,
Zanichelli, Bologna
(1949).
[7] Freedman, M. : "The T o p o l o g y of F o u r - D i m e n s i o n a l Manifolds", Diff. Geom. 17,3 (1982), 357-453.
Jour.
[8] Gieseker, D. : "Global Moduli for Surfaces of General Type", Math. 43 (1977), 233-282.
Inv.
[9] Hartshorne,
R.
: "Algebraic Geometry",
Springer G.T.M.
52,
(I~77).
[10] Hirzebruch, F. : "Topological m e t h o d s in A l g e b r a i c Geometry", Grundlehren 131, Springer-Verlag, H e i d e l b e r g (3rd ed. 1966). [11 ] Kodaira, K. : "On C h a r a c t e r i s t i c Systems of F a m i l i e s of Surfaces with O r d i n a r y S i n g u l a r i t i e s in a P r o j e c t i v e Space", A~er. J. Math., 87 (1965), 227-256. [12] Kodaira, K. - Morrow, J. New York (1971).
: "Complex Manifolds",
Holt-Rinehart-Winston,
[13] Kuranishi, M. "New Proof for the E x i s t e n c e of L o c a l l y C o m p l e t e F a m i l i e s of C o m p l e x Structures", Proc. Conf. Compl. Analysis, Minneapolis, Springer (1965), 142-154. [14] Miyaoka, Y. : "On the Chern Numbers of Surfaces of General Type", Inv. Math. 42 (1977) 225-237. [15] Mostow, G.D. : "Strong R i g i d i t y of L o c a l l y Symmetric Spaces", Ann. of Math. Studies 78, P r i n c e t o n Univ. Press, (1973).
112
[16] Murakami, S. : "A Note on C o h o m o l o g y Groups of H o l o m o r p h i c Line Bundles over a Complex Torus", in "Manifolds and Lie Groups", Progress in Mathematics, Birkh~user, Boston (1981), 301-314. [17] Mumford, D. : "Geometric Invariant Theory", E r g e b n i s s e 34, SpringerVerlag, H e i d e l b e r g (1965). [18] Noether, M. : "Anzahl der M o d u l n einer Classe A l g e b r a i s c h e r Fl~chen", S i t z u n g s b e r i c h t e der Akademie, Berlin, (1888). [19] Severi, F. : "La Serie C a n o n i c a e la Teoria delle Serie P r i n c i p a l i dei Gruppi di Punti sopra una S u p e r f i c i e Algebrica", Comm. Math. Helv. 4(1932), 268-326. [20] Shafarevitch, I.R. : "Basic A l g e b r a i c Geometry", Springer-Verlag, H e i d e l b e r g (1974).
Grundlehren
213,
[21] Van de Ven, A. : "Some Recent Results on Surfaces of General Type", Sem. B o u r b a k i 500, (Feb. 1977), 1-12. [22] Wahl, Jo : "Deformations of Plane Curves with N o d e s and Cusps", Am. J. of Math., 96, (1974), 529-577. [23} Wavrik, J.J. : "Obstructions to the Existence of a Space of Moduli", Global Analysis, Princeton Math. Series 29 (1969), 403-414. [24} Yau, S.T. : "On the Ricci C u r v a t u r e of a C o m p a c t K~hler M a n i f o l d and the C o m p l e x M o n g e - A m p ~ r e Equations, I", Comm. Pure and Appl. Math. 31 (1978), 339-411. [25]
Zariski, O. : "Algebraic Surfaces", E r g e b n i s s e 61, Springer-Verlag, H e i d e l b e r g (1971).
(2nd suppl, ed.) ,
ON A P R O O F OF T O R E L L I ' S
THEOREM
Ciro C i l i b e r t o (~) Istituto
Matematico Universit~
"R. C a c c i o p p o l i " di Napoli
80100 N a p o l i Italia
INTRODUCTION A well known surface
t h e o r e m of Torelli
is d e t e r m i n e d
Torelli's
original
any c h a r a c t e r i s t i c thor s l i g h t l y Schubert's Macdonald
proof
(see
IT])
of the base
enumerative
formulas,
rigourosly
(see
[At])
self
(see
of T o r e l l i ' s
and adapted [A2],
There
is h o w e v e r
to have been
a third p r o o f
ignored
a very e l e g a n t
den in the,
rather
algebraically care which,
longer
proof,
closed
field by G h i o n e
uses
a suitable
ver-
form of R i e m a n n ' s
sin-
given
in 1952 by A n d r e o t t i
by Weil
and A n d r e o t t i
to C o m e s s a t t i
Comessatti's
him-
original
and
exposition.
simple
use of C a s t e l n u o v o - H u m b e r t ' s
(1.8).This s i m p l i f i c a t i o n
of a r e s u l t
(theorem
characteristic
zero.
(*) The A u t h o r
is a m e m b e r
This
believed.
present
(9.1))
very
of G.N.S.A.G.A.
somewhat
ideas,
Anyhow zero,
the p r o o f
(see § 3) has b e e n
of C.N.R.
some
has made
avoiding
to prove,
hid-
on an
required
in C o m e s s a t t i ' s
easy
an ac-
We have w o r k e d
in c h a r a c t e r i s t i c
theorem,
both
of the i n v o l v e d
than we o r i g i n a l l y
in our
to give
(see § 3) b a s i c a l l y
(2.5)),
field of any characteristic. of the p l a i n e s s
[C]), w h i c h
It is,
it useful
arg~aent
(1.8)
(see
literature.
and we b e l i e v e
(theorems
strictkingly
and in our t h e o r e m
due
in the c u r r e n t
obscure,
closed
in spite
the e x p o s i t i o n can be made
t h e o r e m was
to any c h a r a d t e r i s t i c
upon two results
by virtue
later over • by
[W]).
count of it in this note. relies
only
proof
and a weak
[M]). This Au-
theorem.
A second proof
opinion,
theorem
to
the use of some
proved
and over any a l g e b r a i c a l l y
Riemann variety.
and adapted
(see
avoiding
On the other h a n d M a t s u s a k a ' s
sion of C a s t e l n u o v o - H u m b e r t ' s
seems
field by M a t s u s a k a argument
[MC]),
compact
jacobian
has b e e n r e v i s i t e d
Torelli's
(see
that any
polarized
modified
(unpublished).
gularity
states
by its p r i n c i p a l l y
the
argument achieved
but only
in
114
§ I. - SOME
SUBVARIETIES
LINEAR
d-th
field
a complete, k.
symmetric Let
For
non-singular
any p o s i t i v e
product
Any
(r + 1 ) - d i m e n s i o n a l
ear
series
shall
formed
the
If d ~ r, it is C ( d , g ~ )
cases,
(1.1)
all d i v i s o r s
f r o m n o w on we
over
d, we
shall
first
OF A C U R V E
RELATED
TO
an a l g e b r a i c a l l y
shall
one,
indicate
D E C(d)
denote
class
by C(d)
by
by C(d,g~)
thus
d > r. We
of d e g r e e
corresponds
denoted
contained
= C(d);
assume
Chern
of H°(C,i)
complete
of C(d)
instead
integer
subspace
in the
d ~ n, we
by
curve
on C w i t h
vector
g~ c o n t a i n e d
integer
g~.
PRODUCT
the
of C.
L be a line b u n d l e
positive
SYMMETRIC
SERIES
L e t C be closed
OF THE
any
closed
subset
in some
shall
to a lin-
]ii . For
the
in o r d e r
n.
divisor
to a v o i d
also w r i t e
of
trivial C(d,i)
of C(d, I L l).
Lemma.
C(d,g~)
Proof.
is a pure
L e t V C H°(C,L)
r-dimensional
be
the v e c t o r
gnr and
let S o , . . . , s r be a b a s i s r to C ( d , g n) if and o n l y if
c l o s e d subset of C(d).
space
of V. A d i v i s o r
corresponding D = Pl + ' ' "
to the
+Pd
belongs
< r rankllsi (pj) lli=0, ...,r --
(1.2)
j=1,...,d where, the
the q u e s t i o n
field
k. F r o m
at l e a s t
has
Remark.
structure
(1.2)
dimension
of C(d,g~)
(1.3)
being
local,
si(P j)
it f o l l o w s
r. On the o t h e r
at m o s t
dimension
The p r o o f
of scheme,
of
be v i e w e d
any
component
hand,
it is c l e a r
r. W h e n c e
lemma
b u t we
that
can
(1.1)
shall
of C(d,g~) that
of has
any
component
C ( d , g nr) has
a natural
the a s s e r t i o n .
shows
not be
as e l e m e n t s
that
interested
in it in w h a t
fol-
lows. Let
gr be n
a linear
series
without
4: C ~ be
a morphism
determined
non-degenerate ~
C ~ A be
rable,
there
curve
the
by
in ~ r .
unique
gr,
is a c o m m u t a t i v e
F = ~(C)
such
diagram,
being
F is
on C,
and
that
~ = ~ o ~.
uniquely
)
~'
c
a complete,
the n o r m a l i z a t i o n
morphisms C
points
let
F C ~r
If ~: A ~
morphism
base
A
irreducible, of F,
If ~ is not
determined
let
sepa-
up to iso-
115
where
~i is p u r e l y
phisms,
and
inseparable,
9s is i n s e p a r a b l e .
hence
a composition
~s = ~ 0 ~s:
Any
of g r is a m u l t i p l e of a d i v i s o r r 9 = ~s' g r = (gn)s , a n d w e s h a l l
we put
(1.4)
Remark.
strictly
If g r is n o t
contained
of d i m e n s i o n
separable,
in a multiple
Let gr be a separable
of
linear
r o n C, of
denoted
(gr) s.
say t h a t
i t is n o t
( r gn)s •
by
If 9 is s e p a r a r g n is separable.
complete.
In f a c t g r is
(gr) s
series.
It is
to be composite
said
with an involution o f d e g r e e m > I if t h e r e e x i s t s a c o m p l e t e , singular
curve
C'
mor-
r _ C ]?r
to a l i n e a r
divisor
series
C ~
corresponds
ble,
of F r o b e n i u s
The morphism
and a separable,
dominant
non-
morphism
f: C ~ C' of d e g r e e o n C'.
m such
Any
g [ is the p u l l - b a c k ,
g~, w i t h o u t
an i n v o l u t i o n With
that
the
if
(g nr ) s
above
base
points,
will
via
f, of a l i n e a r
be s a i d
to b e c o m p o s i t e
g~.
notation,
Given
g~r
with
is so. the
triple
(C,C',f)
is s a i d to be an inv~
lution o f d e g r e e m > I o n C. R e m a r k t h a t if C' ~ p 1 separable
series
an involution
(C,C',f)
i t is in f a c t
of d e g r e e
m,
a natural
a mor-
phism F: C' ~ C(m) arises,
associating
to e a c h p o i n t
Q @ C'
the
geometric
fibre
Q.
F m a p s C' b i r a t i o n a l l y o n t o a c u r v e d e n o t e d b y x ~ ( C , C ' , f ) , I b y Y m if t h e r e is n o c o n f u s i o n . It is f a i r l y o b v i o u s that: (i) X mI is n o t c o n t a i n e d i n the d i a g o n a l (ii) for a n y p o i n t P E C the d i v i s o r C(m,P) cuts (1.5) then
Remark.
single
If C' C C(m)
for a n y P 6 C, C ( m , P )
at their C',
y1m i n one
common
so t h a t
point.
all P . ' s
of f o v e r or s i m p l y
of C(m) ;
= {m E C(m) : D > P}
point
I Xm(P)
is a c u r v e
(namely F(f(P))). enjoying
a n d C' h a v e
In f a c t
the p r o p e r t i e s
multiplicity
let D = PI + "'" + P m
are d i s t i n c t .
Then
all C ( m , P )
(i) , (ii),
of i n t e r s e c t i o n be
a generic
c u t C'
at D,
one
p o i n t of a n d one,
1
at
least,
sertion
transversally,
follows
equivalent morphism
observing
divisors
of d e g r e e
since that
o n C(m). m
a l l C ( m , P i)
are
{ C ( m , P ) } p E c is
Hence
C'
transversal a system
is n o n - s i n g u l a r
at D.
The
as
of algebraically and the dominant
116
f: p E C ~ C(m,P) gives
rise
ticular mines
to an i n v o l u t i o n
(C,C',f)
F is an i s o m o r p h i s m
N C' E C' such
b e t w e e n C'
that C' = X 1 ( C , C ' , f ) . In parim and y1m, and Ym c o m p l e t e l y d e t e r -
the i n v o l u t i o n . G~ven
an i n v o l u t i o n
I Y m on C and a p o s i t i v e
i n t e g e r d < m, we d e n o t e
by C ( d , y _~) the c l o s e d s u b s e t of C(d) c o n s i s t i n g of all d i v i s o r s D E C ( d ) I such t h a t D E Xm" C l e a r l y all i r r e d u c i b l e c o m p o n e n t s of C(d,x1m) have dimension If
one.
(C,C',f),
(C,C",g)
is s a i d to be c o m p o s i t e (C",C',h)
such
(1.6) Lemma.
are d i s t i n c t
with
Let
a common
a third
one
(C,C',f),
component,
of degree
Proof.
(C,C",g)
then
the u n i q u e p o i n t
(see r e m a r k
(1.5)).
I
involutions
Yn and
C(~,X1n) a n d C(£, X )
that
two i n v o l u t i o n
an b o t h
integer
composite
with
such that C(d,X1n ) and C(d,y1m )
c o m p o n e n t and let X be t h i s c o m p o n e n t .
ynl (p) and yml (p) is a d i v i s o r
with
two d i s t i n c t
d >_ ~.
p o i n t of C, by the d e f i n i t i o n with
be
~ > I such
the
L e t d be the m a x i m u m
have a common
on C, tne f i r s t one
is an i n v o l u t i o n
that f = h 0 g.
Y m an C. I f t h e r e is an i n t e g e r have
involutions
the l a t t e r if there
of d the g r e a t e s t of d e g r e e
If P is the g e n e r i c
common divisor
d on C w h i c h
exactly
of
coincides
cut out by C(d,P)
on y. W h e n c e y is a Yd on C to see t h a t b o t h Yn1 and YmI are c o m p o s i t e
It is e a s y
I
this Yd"
(1.7) Remark.
We e x p l i c i t e l y
C(d,x1n) and C(d,y1m) (1.5)).
Further,
t h i r d one, implies
completely
out t h at the c o m m o n
determines
and C(2,y1m ) h a v e
for P g e n e r i c on C,
component
the i n v o l u t i o n
if y1n and X mI are two i n v o l u t i o n s
t h e n C(2,y1n)
that,
point
YdI (see r e m a r k
not c o m p o s i t e
no c o m m o n
the g r e a t e s t
y of
component.
common divisor
with
a
This of yn(P)
and X mI (p) is just P We are now able (1.8) T h e o r e m . out b a s e
points
Let
to p r o v e
r gnl,
on C.
C ( d , g r I) a n d C ( d , g r 2 ) both
composite
Proof.
with
the
r be two d i s t i n c t gn2
If there have
is an i n t e g e r
a common
the s a m e
d > r + I such
component,
then
is a p a r t i c u l a r
two s e p a r a b l e
~i: C ~ F i _C determined
linear
the
two
series
with
that series
are
involution.
If r = I the s t a t e m e n t
L e t n o w r _> 2. We h a v e
complete
r , i = I ,2; let by gn~
1Dr
morphisms
case of lemma
(see r e m a r k
i = 1,2
(1.4))
(1.6).
117
~i: be
the n o r m a l i z a t i o n s
F
,
i
i = 1,2
and
Ai
of F., l
~i: be
the m o r p h i s m s
C and
such
that
let us c o n s i d e r
the
gr (p) = ni These
are
linear
series
i
1,2
=
let
C ~ Ai
~i = zi 0 ~i' linear
i = 1,2.
L e t P be
any p o i n t
{D E C ( n i - 1 ) : of d i m e n s i o n
D +P r -1
6 g~} , i
i = 1,2
> I and d e g r e e
n.-
--
m a y be,
fixed
geometric and DI(P), and
divisors
fibre D2(P)
remark
composite series
the
case
subset
~I'
assume
we h a v e
with
chose
get
i = 1,2, the
a not empty
Zariski
consequence
pendix,
lemma
(1.9)
Remark.
to r e m o v e
assertion.
A,B
points
Let
open
since
case,
In case
the
and
are
is n o t
above,
of k is
of c o m p l e t e n e s s to s i m p l i f y
If we
= ~I(B)
and
containing
rational. (see
P on C,
composite
I IP)
we with
zero
Then
Then
the p r o o f
P in it is
[M], A p -
y~(P)
is in-
get a c o n t r a d i c 1 this yZ.
it is not
for the
F2
a n d we
for any p o i n t
theorem
points
like
~I(A)
involution, that
=
involution,
is i m p o s s i b l e .
assume
many
reasoning
is p o s s i b l e
g 2 (Pl
in the ~ r
the s a m e
of C, y~(P)
Assume r-1(p) are gni
P on C.
same
that
line A'B'
characteristic
the h y p o t h e s i s
points
on C such
with
i = 1,2,
y~(P). Since
i = 1,2. This is impos r gn2 are d i s t i n c t . So we
the
of F2, w h i c h
for i n f i n i t e l y
f r o m g~. (P) we l By i n d u c t i o n we may a s
t > I. O f c o u r s e many
the
points
P on C.
g~1'
with
(1.6)
So let us e x a m i n e
involution
points
us f i n a l l y
subset
also
lemma
for P in a n o t e m p t y
I (r-1)y~(P)l,
on the
composite
(C,Ai,~i),
(1.8),
composite
of the C a s t e l n u o v o - H u m b e r t
In this
both
ficul
in t h e o r e m
Then
many are
3) that
of P.
unless
are b o t h
are
composite
points
assertion.
an e a s y
dependent
not
= B'.
lie i n f i n i t e l y
(C,Ai,~i),
i = 1,2,
YtI for i n f i n i t e l y
were
two d i s t i n c t ~ ~2(B)
(C,Ai,~i),
divisors,
with
on C
involution
the same
birational,
the
by
the
many
to be a y~,
this
i = 1,2,
= ~2(A)
with
is j u s t
if P is g e n e r i c divisor,
eliminating base r-1 gni (P), i = 1,2.
gnir-1(P) =
~2 are b o t h
is c o m p o s i t e
get
for i n f i n i t e l y
(C,AI,~I)
(C,Ai,~i),
series
composite
is r a t i o n a l
if
of C. Then,
linear
are b o t h
series,
tion
this
and we
sible
again
With
P +Di(P)
Thus
common
involutions
one.
are d i s j o i n t
complete
would
the two
Clearly
greatest
i = 1,2,
they
could
a non-zero
I, with,
1
i = 1,2.
i = 1,2,
complete
A'
P,
Di(P),
get two
may
i = 1,2.
r gni,
open
y~(P)
have
a third
Zariski
sume
Di(P),
~i c o n t a i n i n g
(1.7), with
two
of
of
series
dif-
two s e r i e s in s o m e
g~i
points.
118
§ 2. - E L E M E N T A R Y We
assume,
variety
o f C.
PROPERTIES
OF THE JACOBIAN
f r o m n o w on,
For
C of genus
any positive
integer
~d: C(d) be
the A b e l - J a c o b i
C. We put,
map,
defined
g ~ d,
VARIETY
OF A CURVE
1. L e t J(C)
be
the
jacobian
let
~ J(C)
with
respect
to a f i x e d b a s e
point
on
as u s u a l W d = 9d(C(d)),
0 = Wg_1
~ 2 g - 2 (Kc) = k where
K
is a n e f f e c t i v e c a n o n i c a l c d e n o t e b y T v the translation
shall
u E J(c)
divisor
~ u+v
on C.
For
any v E J(C)
we
E J(C)
and by T v ~ the reflection u E J(C) Since
~ -u +v
~ J(C)
clearly o = Tk(O)
it is T:v(@ ) : T v _ k ( Q ) for a n y v E J(C).
divisor
a theta divisor.
called where
Any
~g, w h i c h
on J(C)
We a l s o p u t
is a b i r a t i o n a l
(2.1)
o f the
SC(g)
morphism,
type
Tv(0),
v E J(C)
is
= C(g, I K c l ) : t h i s is t h e
locus
fails
If
to be i n j e c t i v e .
W = ~g(SC(g)) we have W =
Hence
W, Let
2g-I. (2.3)
as w e l l now
Clearly Lemma.
ible divisor (ii) if (iii)
i be
as W g _ 2 , any
(2.2)
is i r r e d u c i b l e .
line b u n d l e
it is C ( g , i ) (i) If
T ~ (Wg_2)
on C w i t h
~ SC(g).
Further
I[I has no base points,
first Chern we have
class
of degree
the
then C(g,i)
is an irreduc-
on C(g);
Ill has a base point P, then C ( g , i )
the image W(L)
= SC(g)
via ~g of the irreducible
O C(g,P);
component
of C(g,i)
not
119
c o n t a i n e d in SC(g) is a theta divisor; (iv) any theta divisor is a W(i). Proof. D E C(g)
If E E C ( g - I )
such that E + D
is general enough,
@
ILl. A c c o r d i n g l y
there is only one
there exists a natural bi-
rational map f: C ( g - I) ..... ~ C ( g , / ) which is d o m i n a n t onto a c o m p o n e n t of C(g,i). and D ~ SC(g),
Further,
let E 6 C ( g - I) be such that D + E
unique divisor such that D + E C f(E) = D. Thus,
6
if D @ C(g,i)
I/I. Then D is the
Ill, namely f is d e f i n e d at E and
if there is any other c o m p o n e n t of C(g,£),
a c o m p o n e n t of SC(g)
too. This implies,
by t h e o r e m
it has to be
(1.8), that
IKel; hence
Ill has
a base P o i n t P. If this happens it is
Ii(-P) I =
(ii) follow. Take now any divisor D E
ILl and put v = ~2g-I (D). Clearly
(i) and
it is W(i) = T ~ (@)
(2.4)
V
Since ~2g-I imply both
is surjective, (iii)
and
v is any point of J(C);
thus
(2.1) and
(2.4)
(iv).
C o n s i d e r now the algebraic family of theta divisors on C, w h i c h is p a r a m e t r i z e d by an i r r e d u c i b l e divisors of J(C);
c o m p o n e n t H of the H i l b e r t scheme of
H is i s o m o r p h i c
to J(C).
Let 0(W) be the closed sub-
set of H formed by points of H c o r r e s p o n d i n g to theta divisors c o n t a i n ing W. We have the (2.5) Theorem.
W(1)
contains W if and only if
Thus @(W) is isomorphic Proof.
ill has a base point.
to C.
If D E C(g,i),
then
IDI = ~gl (~g(D)) C C(g,i).
Hence
W(i) _D W implies C(g,[) and, by lemma
(2.3),
O SC(g)
ILl has a base point.
base point P, again by lemma
Conversely,
if
ILl has a
(2.3), it is
W(i) = 9g(C(g,P)) D W Finally and
the i s o m o r p h i s m b e t w e e n 0(W)
(iv) of lemma
(2.3).
and C follows by virtue of
(iii)
120
§ 3. - C O M E S S A T T I ' S Let We
shall
C' be
another
denote
Assume
PROOF
complete,
anything
there
OF T O R E L L I ' S
non-singular
concerning
exists
0 onto
We may
as w e l l
By
theorem
(3.2)
@'.
Theorem. F(W)
taking
onto
Proof.
The
after
Torelli's
There
curve
an u p p e r
of genus
g over
k.
"prime".
~ J(C')
L e t P E C, P' E C' a n d s e t
assume,
(2.5)
C' by
an i s o m o r p h i s m F: J(C)
taking
THEOREM
is
having
used
i=0c(Kc+P),
translations,
F(W(i))
= W' (i')
theorem
is a c o n s e q u e n c e
either
i' = 0 c ( K c , + P ' ) . that (3.1)
a translation
of the
following in J(C')
or a r e f l e c t i o n
W'. assertion
is
trivially
assume
f r o m n o w on g > 3. S u p p o s e
(2.2),
it is F(W)
the
true
for g = 1,2;
theorem
is n o t
thus
true.
we
Then,
shall by
(3.3)
~ Tv, (W')
(3.4)
F (W) ~ Tv, (Wg-2 ) for any
v' E J(C').
sional,
closed
By
subset
(3.3),
there
exists
A of C' (g), w h i c h
(g-2) - d i m e n
an i r r e d u c i b l e ,
dominates
F(W)
via
~g,
and
such
that A ~ SC' (g) Lemma
(2.3),
(3.1)
and
(3.5)
imply
(3.5)
that
A C C' (g,P') Assume
now
a linear
another
series
A _C C' (g,P~).
theta
with
Thus,
divisor
a base by
point
(3.7),
W' (i~) P~
in
A is i r r e d u c i b l e
(3.7)
Hence
the e q u a l i t y
we may
suppose
and
the r i g h t
holds.
This
I[41 w i t h o u t B =
containing
# P'.
Then,
F(W)
like
corresponds
above,
we
to
get
it is
A _C C' (g,P') Since
(3.6)
A C' (g,P~) hand
side
is e a s i l y base
is i s o m o r p h i c
seen
points.
{D E C ' ( g - 1 ) :
(3.7)
D+P'
to C' (g-2),
to c o n t r a d i c t
We p u t eA}
(3.4).
121
B is a c l o s e d
s u b s e t of C' ( g - I), i s o m o r p h i c F(W)
where
p' = ~I(P').
Now
(3.6)
to A. M o r e o v e r
it is
I = Tp, (9g-I (B))
(3.8)
implies
B C C' (g-l,
L~ (-P'))
g-2 has an e f f e c t i v e d i v i s o r M of d e g r e e h > 0 of b a s e if IL~(-P')I : g2g-2 points,
there
are two cases
to be c o n s i d e r e d :
(i) there is a p o i n t Q' in M s u c h t h a t B C C' (g-l,
(ii) Case
B C C' (g-l, (i) leads
account
(3.8).
Q')
L4 (-P'-M)).
to a c o n t r a d i c t i o n , Case
argueing
like above
(ii) is a l s o i m p o s s i b l e . linear
chosen
10c, (Kc,) ~ i~ (-P-M)~[.
F2
would
in a d i v i s o r
of
series.
In fact
= g g 2-h is a s p e c i a l
Therefore
and t a k i n g
Ii~ (-P-M)I
into
=
a p o i n t R' can be So if v' = ~4 (R'), it
be T v, (9'g_i (B)) _C W
contradicting, T a k e n o w any with
by
(3.8),
(3.3).
L~, d i f f e r e n t
i~ ~ L~.
In force of t h e o r e m
f r o m i', s u c h that W' (i~) D F(W),
i = 1,2.
L~(-P'))
nC'(g-1, _
g-2
y',
linear series
m > I. L e t
a > g-2,
we
the o n l y p o s s i b l e
IL"(-P')]
involution,and
g g 2 (i) is p u l l - b a c k
F2
via f
L~ = L~. Moreover,
values
for m are 2,3,4. similarly
If m = 2, we s h o u l d h a ve for any
w o u l d be c o m p o s i t e (3.9)
2
<2
a = I and i~ ~ i½, a c o n t r a d i c t i o n ; a = 2, and
lows by
are c o m p o s i t e
get
~ <
e l l i p t ic.
(3.9)
It is
since
whence
(C,C",f)bethis
on C" of w h i c h
am = 2gand,
i = 1,2,
L~(-P'))
(1.8), ,iIL'(-P)I~ - g2g_2 (1), i = 1,2,
the same i n v o l u t i o n
gag-2 (i) the
Ii~ (-P') I has no base points.
It is B_CC'(g-I,
with
Thus
and r e m a r k
But m = 4 i m p l i e s
m = 3 implies a = g-1
g = 4,
and C" w o u l d be
i" ~ L', such that W' (i") D F(W), with
(1.7),
this e l l i p t i c
y~:
for g = 3 this
for g > 3 it is obvious.
This
also
fol
122
leads nal
to a c o n t r a d i c i t o n ,
to C",
against
§ 4. - R E M A R K S
ABOUT
In the p r o o f turn, rem.
we m a d e This
in w h i c h
(4.1)
of
THE
can be
CASE
avoided
(3.2)
we
use
version
the
following
weaker
not composite with an involution,
C(d,g~)
is irreducible.
as the
full
(1.6)
in
For
assertion
symmetric
Theorem ma
theorem
if c h a r k = 0. We w a n t
points,
The
(1.8) § 2,
@(W)
is b i r a t i o -
(1.8),
in w h i c h ,
of C a s t e l n u o v o - H u m b e r t ' s to b r i e f l y
version
If g~, r ~ 2, is a linear series
Theorem.
that
in theo
sketch
here
done.
one p r o v e s
Proof.
imply
char k = 0
theorem
it can be
it w o u l d
(2.5).
use of M a t s u s a k a ' s
way
First
because
theorem
follows
group
can be
on
from
(1.8).
on C, without
base
then for any integer
the
fact
the g e n e r i c
throughly
of t h e o r e m
that
divisor
replaced
by
d ~ r+1,
the m o n o d r o m y
of the
theorem
acts
gr n
(4.1)
and
lem-
3.
instance,
the p r o o f
of
(i) of l e m m a
(2.3),
in w h i c h
theorem
(I 8) w a s e m p l o y e d , f o l l o w s by t h e o r e m (4•I) since if [LI g-1 has • ' ' = g2g-1 no b a s e p o i n t s , it is n o t c o m p o s i t e w i t h an i n v o l u t i o n : this can be checked
by e a s y
Only
the
comments. that
computations•
last
We k e e p
W(L")
_O F(W),
part all
of
the p r o o f
introduced g-2 IL"(-P') I = g2g-2
then
B C C' (g-l,
Since
clearly
(4.2)
the e q u a l i t y
C' (g-l,
of all C' (g-l, all
linear
cheeks
again
y~ b e i n g since
the p r o o f All
m = 2,3,4, is
finitely
above
theorem
deserves
some
L" ~ L' is such
no b a s e
points
and
determines
(4.1),
that
(4.2)
i",
composite
but
a priori many
has
If
i"(-P'))
for any
By t h e o r e m
(3.2)
there.
completely
not hold
IL''(-P') I are
of t h e o r e m the
one p r o v e s separable.
that
elliptic,
C' has
L" (-P'))
does
L"(-P')).
series
of t h e o r e m
notation
or
lemma
with
for
I [''(-p') I , in
and B is a c o m m o n (1.6)
for g = 3,
an i n v o l u t i o n
generic
i" only
possible.
This
'{2 does
automorphism.
Then
the
not
component
X1.
One
the c a s e m = depend
conclusion
is
2,
on
i",
like
in
(3.2).
argument (4.1).
applies
Note
that
to the one has
case
char k > 0, p r o v i d e d
to a s s u m e
at
least
the
r
gn
123
REFERENCES [All
A. ANDREOTTI: res,
[A2]
Acad.
A. ANDREOTTI: (1958),
[c]
[MC]
I.G. logy,
[T]
[w]
On a t h e o r e m
Atti
Sulle
(1962),
Am.
(1952),
irr~guli~ 3-36.
J. of Math.,
80
hermitiane
Torino,
50
of Torelli,
d e l l a variet~
(1914-15), Am.
di
439-455.
J. of Math.,
80
98-103.
(1913),
products
variet~
Z um Beweis
curve,
di Jacobi,
Rend.
R. Acad.
des T o r e l l i s c h e n
Satz,
Nachrichten
der W i s s e n s c h a f t e n 33-53.
of an a l g e b r a i c
Topo-
319-343.
(5)
(1957),
Sci.
Symmetric
Sulle
Akademie
27
784-800.
R. TORELLI:
A. WEIL,
alg~briques
de Sci.,
of Torelli,
On a t h e o r e m
N!ACDONALD: I
les surfaces CI.
trasformazioni
R. Accad.
T. MATSUSAKA: (1958),
sur
de Belgique,
801-821.
A. COMESSATTI: Jacobi,
[M]
Recherches
Roy.
in G~ttingen,
Lincei,
Math.-Phis.
22
der Klasse,
2
TWO E X A M P L E S OF ALGEBRAIC T H R E E F O L D S WHOSE HYPERPLANE SECTIC~'S ARE E N R I Q U E S SURFACES by ALBERTO CONTE D i p a r t i m e n t o di Matematica,
U n i v e r s i t ~ di T o r i n o
Algebraic t h r e e - d i m e n s i o n a l varieties whose h y p e r p l a n e sections are Fnriques surfaces were studied e x t e n s i v e l y in thirties by G O D E A U X and FANO. Recently, J'. P. MURRE and m y s e l f have taken up this subject g i v i n g a p r o o f according to modern standards of FANO's main theorem on this class of threefolds: MAIN T H E O R E M (see IF] and [C-M]).- Let W ~ P ~
be a p r o j e c t i v e l y normal algebraic
threefold such that, for a sufficiently general hyperplane H, the h y p e r p l a n e s e c t i o n F = W.H is a smooth E n r i q u e s surface. Let us denote by p the genus of the curve C = = W.H.H', where H' is a hyperplane different from H. Then, N = p, deg W = 2p - 2 a n d W = W 2p-23 is b o u n d to have a finite number of s i n g u l a r points PI' Assume that W is not a cone
, that p ~ 6 and that the points P. are similar, i
e. that they all have the same properties. P
"" " ' Pn" i.
Then, n = 8 and each o f the points PI . . . . .
is a quadruple point with tangent cone the cone over a V e r o n e s e surface. Moreover, 8
W carries a linear system
I~I of Weil divisors the general m e m b e r of w h i c h is a K 3
surface. This system has dimension p - i and the base points of it are the P.'s, each 1 - o f w h i c h is a rationale double p o i n t on a g e n e r a l ¢ . Let__ .: ~I~I: J
be the rational map
defined by
w
-
-
-,
M
~ e p-I
i
I~I. Then,
~I~I
is always birational and M =
k 1el(W)
spans a pp-l, has degree 2p - 6 and has K3 surfaces as (general) h y p e r p l a n e sections (i. e. is a "Fano threefold" in the classical sense). Furthermore, M contains 8 planes ~i'
"''' ~ 8 w h i c h are the images of the points PI'
s i n g u l a r i t i e s in the points common to two of the
;
"''' P8 and has at most isolated 's.
125
We will assume that no three o f the
~ 's meet in the same point. One can show i that ~ and ~ meet if and only if the line P P lies on W. In this case the points i j i O P
i
and P
j
will be called "associated".
FANO
claims that such threefolds exist only for p = 6, 7, 9, 13 (plus the excep-
tional case p = 4, g i v i n g rise to the s o - c a l l e d "Enriques threefold). sible that a few more cases
exist.
It is h o w e v e r pos-
In this paper we will give a geometric d e s c r i p t i o n
of the two most i n t e r e s t i n g cases, c o r r e s p o n d i n g to the values p = 6 and p = 9. For more details and complete proofs, see the second p a r t o f LJ[C-M] and [C-V].
I.- The case p = 6 Here the variety W = W
i0
6 is of degree i0 and lies in P . By p r o j e c t i n g W from the
3 plane o f any three of the P ' s 1 sociated,
so that the planes
3 onto P , one sees that the P.'s must be two by two as1 ~
's will m e e t two by two in one point. i 5 The corre.~ponding M = M 6 will be o f degree 6 will lie in P and will have canoni3 cal curve sections of genus 4, so that, form the c l a s s i f i c a t i o n of Fano threefolds, M = 5 = Q'C will be the intersection of a quadric and a cubic h y p e r s u r f a c e s o f P
containing
8 planes two by two incident. One can show that Q is a n o n - s i n g u l a r quadric, so that , in order to describe M and its geometry,
it is c o n v e n i e n t to identify Q w i t h the G r a s s m a n n i a n G = G(I,3) of
lines o f p3 (so that M will be what is c l a s s i c a l l y called a cubic complex of lines)• Remember now that a base for the integral homology ring of G is g i v e n by the Schubert cycles
(where p , 1 , h are r e s p e c t i v e l y a fixed point, o o o
line and plane o f p3):
~l(lo) ={x'G I lxnlo~} %(Po ) = lx,G I ix ' Po} ~l,l(ho ) = {x ~G g2,1(Po,ho)
I ix ~ ho }
: {x'G
I PO ' IXC- ho 1
w i t h the f o l l o w i n g intersection relations: 2 gl
= ~2 +
~2 2
~i ,i '
~i'%
~ i,i = ~ 2 ,i '
= ~I"
2 = ~ i,i
=~i
~
2,1 = i,
~
2
~
i,i = O.
3 Let now R be a n e t (i. e. a 2 - d i m e n s i o n a l linear system) of quadrics of P . It is well k n o w n
that:
126
V (R) = { x • G I 1 0
lies on some quadric Q of R } X
is a cubic complex. On the other hand, M too, as we have seen,
is a cubic complex and
the 8 planes on it, m e e t i n g two by two, will be of the kind ~ 2 ( P l ), ..., ~ 2 ( P 8 ), whe3 re the Pi'S are points in P . Moreover, through the seven points PI' "''' P7 there will certainly pass a net of quadrics R and, if O is a quadric of R not c o n t a i n i n g any of the lines p.p., then, r e m e m b e r i n g that: i 3 V(Q) = { x ~ G I 1 Q }= 4 ~ = c + c (two conics, c o r r e s p o n d i n g to the two x 2,1 1 2 rulings of Q), one should have, on one hand: M'c
=
6,
I whilst, on the other hand: M'c I ~ 7 (the seven lines of the rulinz of Q g o i n g through PI'
"''' [7 )'
so that M = Vo(R) and R goes also through P8" We have ther'efore proved the following: 5 = Q'C ~ P c o n t a i n i n g 8 planes with the configura3 3 there exists a net of quadrics R o f P such that M = V (R). o
P R O P O S I T I O N i.- For any M = M tion considered,
6
P R O P O S I T I O N 2.- M is rational. Proof.- Let R = P
2
and let D be the plane quartic curve
(Hesse curve) correspon-
d i n g to the cones of R. Let S be the rational double plane b r a n c h e d over D. Let's define a rational map: : M - - -) S by sending the lines of c to Q
and c r e s p e c t i v e l y into the 2 points of S c o r r e s p o n d i n g I,Q 2,Q S. D is the b r a n c h locus because the two rulings coincide for the cones. The fi-
bres of ~
are the conics Ci,Q, so that M is a conic bundle over the r a t i o n a l surface
S. Moreover, g2(Pj)
any of the planes g2(Pj)
• ci,
Q
=
1 point
is a rational surface such that:
(the line of ci, Q g o i n g through pj
~
Q).
Therefore, M is rational by the E n r i q u e s criterion of rationality. Let now P be a pencil of quadrics c o n t a i n e d into R. The base locus o f
P - the
i n e t r s e c t i o n of two quadrics s p a n n i n g P - is a quartic e l l i p t i c space curve and: V (P) = I x ~ G I 1 0
lies on some quadric of P 1 X
is n o t h i n g else than the (2,6)-congruence of the chords o f C, i. e.:
127 V (P)~
2 o
+ 6 o
•
o 2 i,i Let now R' be any net other than R going through P. Then: 2
V (R).V (R') ~ o o = V
(P)
9 ~
= 9 ~2 + 9 ~
1
= (2 ~2 + 6 ~i,i ) + (7 ~ 2 + 3 ~i,i ) =
i,i
+ F,
0
where F ~ 7 ~ 2
+ 3 o
PROPOSITION
1,1
3.- F is an Enriques surface•
Proof.- The two nets R and R', having a common pencil, F = {X~ G I 1
lies in all quadrics
will span a web S and:
of a pencil P' C_ S }
x
(since 1
lies on one quadric of R and one of R' ). Therefore, x ce, called classically "Reye congruence" (see [B], p. 136). We have deg F = 7 + 3 = i0 and dim ing through P) so that
IF1 = 6 (since there are O0
6
surfa-
nets R' ~ R go-
IFI will define a rational map: 6 _ _ -) w I O c
XIF!: To identify W = W I0 3
3
F is an Enriques
v9
S3v9
let
P
p6
3 --
"
be the projective
space parametrising
the dual
'
quadrics of P . Inside P
there is a well known filtration given by the rank: ~9 P Ul A84 = {quadrics
of rank _<3 1
UI i0 3 V 6 = {couples of points of P } Ul v8 3 = {points 8 V lies 3
Moreover,
inside
10
V 6
L e t u s now r e m e m b e r t h e
DEFINITION,- A quadric
~aijuiu
with
multiplicity
following of
equation
j = 0 are siad to be apolar
Baij
of p3 counted twice }
4.
classical: ~
a
x x
ij i j
= 0 and a dual
quadric
of
equation
if:
aij = O.
It is easy to see that, given a linear system S of quadrics of dimension h, the dual quadrics which are apolar to all quadrics of S make up a linear system S of dimension 8 - h. Moreover, the "couples of points" belonging to S are exactly the couples 3 of points of P which are conjugate (i. e. belong each to the polar plane of the other) with respect to all quadrics
of S.
128
6
v = V (R), let R be the 6-dimensional linear system o f dual quao
Given now M = M 3
drics which are apolar to all quadrics of R. V
P R O P O S I T I O N 4.- WI03 = klFI(M~) = vIO'R'6 Proof.- It is enough to remark that, by a well k n o w n p r o p e r t y of the Reye congru-3 ence, a line 1 ~ F if and only if 1 = xy, where x and y are points of P w h i c h are conjugate with respect to all quadrics of the web S C R, so that (x,y) ~ ~ ~
v
~
R and
S .W3
10
i0 is a h y p e r p l a n e section of W 3 , since dim S = 5. Note that W
i0 3
has 8 quadruple points in the intersection V
8 3
"R.
2.- The ease p = 9 Here the variety W = W
16 3
has degree 12 and lies in
9 12 has degree 16 and lies in P . The c o r r e s p o n d i n g M = M 3 8 P . As to the c o n f i g u r a t i o n of the 8 planes lying on M
one can show the following: PROPOSITIOV' 4.- The eight planes lying on M can be divided in two d i s j o i n t sets ~i' a2' a3' a 4
'
a i (~ aj =
~ i' ~2' ~3' ~ 4
~ i f] ~ j = ~'
such that: a i £] ~j = 1 point.
Let now G = G(2,8) be the g r g s s m a n n i a n o f 2-planes in P ~. : { x , G
I dim(;
1
Each
r h ~ . ) >_0}, X
8
and:
i : i, 2, 3, 4.
1
4. is an irreducible Schubert cycle of c o d i m e n s i o n 4 in G (dim G = 18), so that: 1
4
ra:
N
i=l i is a 2-dimensional cycle in G. P R O P O S I T I O N 5.-
8
(i) If no three of the
a . 's are included in a h y p e r p l a n e of P , then
2 T a
~
P
.
1
(ii) Let:
Va
LJ X~
Then, V a
~
TO,
_C p8. X
is an irreducible n o n - s i n g u l a r algebraic v a r i e t y o f d i m e n s i o n 4 and de-
gree 6. (iii) U n d e r the above hypotheses: Va 2 where
g:
P
2 x P
= V
=
g (p2 x p2),
~8 ~ P is the Segre embedding.
129
Let now M = M 12 be a Fano variety containing 8 planes having the above configu3 ration. Then: PROPOSITION
6.-
(i) For every
m
C V a , M (~ is a curve. x -x (ii) There exists a quadric hypersurface Q c
p8 such that M = Q N V
Proof.- The proof of (i) is rather long and complicated ties of the K3 surfaces which are the hyperplane
and relies upon proper-
sections of M.
From (i) it follows that M C V a = V, so that, 2 2 (p,q) on P x P . Since:
in particular,
M must be of type
12 = deg M = ~(p + q) deg V = 3(p + q), the only possibilities (0,4),
(1,3),
for (p,q) are:
(2,2).
The first is ruled out because M intersects because M intersects (2,2),
both families of planes on V, the second
them in points which are not all on a line, so that M is of type
i. e. M = Q C] V.
It is moreover possible to construct QI:
AI(Yo'YI'Y2)
+ BI(Y3'Y4'Ys)
= 0
Q2:
A2(Yo'YI'Y2)
+ B2(Y3'Y4'Y5)
= 0
such that,
two quadrics of P
5
with equations
of type:
if Y = QI'Q2 and X = Y/i, where i is the involution defined by:
i [(Yo . . . . , y5 )] = (-yo,-Yl,-y2,Y3,Y4,Y5) , there are two suitable
embeddings:
/',, X
9
p Here the
hyperplane
sections
16 ~ W3 . . . .
12 p8. ~ M3 C_
of
W 16 are Enriques surfaces (see [B], pp. 135-36) 3 5 obtained modulo i from the intersection QI'Q2.Q3, where Q3 is a third quadric of P of equation: Q3:
A3(Yo'YI'Y2)
+ B3(Y3'Y4'Y5)"
REFERENCES
[B]
A. BEAUVILLE,
Surfaces alg~briques
complexes,
Ast~risque
n. 54, Soci~t& Math~-
130
matique de France 1978.
[c-M]
A.CONTE and J. P. MURRE, T h r e e - d i m e n s i o n a l al~ebraic varieties w h o s e h y p e r p l a n e sections are Enriques surfaces,
c-v]
Institut Mittag-Leffler,
Report n. i0,1981. 9 A. CONTE and A. VERRA, V a r i e t & algebriche t r i d i m e n s i o n a l i ~ m e r s e in P le cui sezioni iperpiane sono superficie di Enriques,
[F]
to appear.
G. FANO, Sulle v a r i e t ~ algebriche a tre dimensioni le cui sezioni iperpiane soa no superficie di $enere zero e b i g e n e r e uno, Mem. Soc. It. XL, s. 3 , t. XXIV
(1938), 41-66.
On the Brill-Noether
Theorem
D. Eisenbud and J. Harris
The purpose of this note is to give a short, self-contained Brill-Noether
theorem:
Theorem
Let
(i):
C
be a general curve of genus
esses a linear system of degree
d
g , and suppose that
and dimension
0 = g - (r+l)(g-d+r)
proof of the
r .
earlicr in [K-L I],
~ 0 .
the approach here will be to study the behavior
of linear series on a family of curves degenerating
(2)
the converse was
[K-L II] and [K].
As with all existing proofs,
curve.
poss-
Then
This was originally proved in [G-H], and more recently in [E-HI; established
C
to a singular and/or reducible
We introduce our family here:
Notational
conventions:
For the remainder
of this paper,
crete valuation ring with parameter
t , T = Spec 0
the closed and generic points of
respectively.
T
projective family with total space
X
0
will be a dis-
its spectrum, ~ : X-->
T
smooth, and central fiber
and
0
and
will be a flat, X 0 = z-l(0)
the
reduced curve pictured in fig. i.
Our object is to prove theorem fiber
X~ = X xT Spec k(~)
of
(i) specifically
X ; since families
[W]), and since the non-existence
for the geometric general
X
exist for all ~cnera
this will suffice to prove Theorem
first observe that any line bundle
on
k(n)
.
(see
of linear series of given degree and dimension is
an open condition among smooth curves,
sion of
g
L
~
(i).
We
is defined over some finite exten-
But if we make any finite base change
T' --> T
and minimally
132
¥~
ps
Ycd
JEe
;
'~C 2 ~ I
J
i I
I
l
J m~ f'f'~v ]¥%-,
p. YN
Components are smooth, and intersect transversally as shown. The E i are ellipFig. I
tic; all others are rational.
133
resolve the singularities
of
family of the same form as Moreover,
X' = X xT T' , we find that X ; thus we may assume
since the total space of
to one on
X .
Thus, Theorem
X-->
(3):
let
be the relative degree of
P = g - (r+l)(g-d+r)
To prove follows:
T
L
ponents of
on
Y
X
X0 .
L
and
L
(~,Ly)~k(~) ditions;
X0 .
r+l = rank(~,L)
of
X0
pairing among components
.
and such that We define
Vy
Since the
each
Vy
, it is reasonable
of
Ly
has degree
d
on
sequence
we let
c H0(Xo,Ly)
is unimodular, Ly
on
X
agree-
on all other com-
c H0(y,Ly)
,
Ly
vanishing
on
Y
ao(Vy,p)<...~ar(Vy, p)
p£ (cf. fig. i).
(4)
(i)
£
con-
those conditions Theorem 3') will follow
of sections
a~ < "'" < a£r
i ,
as
to expect that they satisfy some compatibility
at the point
and
X
are all limits of the same linear series
Vy~
For all
X ;
to be the linear series
distinct orders of vanishing £ = 2,...,N
X0
Y , 0
These conditions may be expressed as follows:
we define the vanishing (r+l)
extends
Then
there exists a unique line bundle
and indeed, once we establish
immediately.
.
be any line bundle on
the last inclusion coming from the fact that any section of on
X~
the limiting behavior of a linear series on
V y = (~,Ly) O k ( O )
vanishes
k(D)
~ 0 .
since the intersection
ing with
is defined over
any llne bundle on
be as in (2), and let
(3) we consider
for each component
is smooth,
is again a
(i) will follow once we establish
Theorem d
Let
X
L
X' --> T'
~ c Vy
at
p
p £ Y ,
of
Vy
at
p ; in particular,
be the vanishing
Our basic condition
for any point
to be the for
sequence of the series
is then
134
£+i
ai
(ii)
If
£ = c. 3
for some
£
Z a i ; and
j , then for all but at most one v a l u e of
a.
> a.
1
l
.
We note that T h e o r e m (3) follows immediately from (4): for any
i ,
trivially, we have,
£ , i ~ a~ S d-r+i , so that a l t o g e t h e r 1
N
2
(r+l)(d-r) -> E a . - a. 1 1 i a.£ + i - a £ i,£ i -> rg
and h e n c e
p = ( r + l ) ( d - r ) - rg ~ 0 .
We b e g i n the proof of (4) w i t h two lemmas. Y, Z
of
X 0 , m e e t i n g at a point
In this situation,
X O - {p}
p
E
the divisor on
p'
another point of
Y , as in Fig. 2.
has two connected components; we w i l l
¥
denote by
with
Both refer to a pair of components
z
X
consisting of the sum of the curves in
the connected component containing
Z .
In particular, we have then
X0
in
135
L Z ~ Ly(-dE)
we accordingly regard ~,Ly .
LZ
as a subsheaf of
Finally, for any element
conventions,
Lemma 5.
i) and ii)
Proof:
and
z,L Z
we will write
section of
Ly
There exists a basis
~O,...,Or
for suitable integers
~-i >- 0
the orders
ordp, y(O i)
of
~,Ly
0
along
P,y
(~)
for the
Y . With these
the set
t iO i
~,L Z
are all distinct.
0-modules
~0,...,Or
of
Z,Ly
satisfying
a i ~ ~j , then i) will still hold if we replace
transformations
is a basis for
n,Lz-->
Z,Ly
may
by applying Gaussian elimination to its rows and columns;
this procedure yields a basis
If
ord
such that
The matrix expressing the inclusion of free
Lemma 6.
as a submodule of
then, we have
be diagonalized over
and
Ly
O ~ ~,Ly
order of vanishing of the corresponding
;
~i
suffice for passing to a basis satisfying
O ~ ~,Ly- t.n,Ly
and
by
(i).
Now, if
g £ 0
oi+ g~j ; these
(ii) as well.
T = Ta'O ~ z,L Z- tz,L Z . then we have
ordp,,y(O ) ~ d- ordp. y(O) ~ ~ ~ ord p,Z (T)
Proof:
The first inequality is trivial (but is the key to (7)(iii) below).
the second inequality,
observe that since
~Xo+ (o)
thus
(g) e (d-a)E
=
and correspondingly
inequality we see that since
For
T ~ ~ z,L Z , the divisor
(t~O) Z dE ;
ordp,y(O) ~ d-a .
t-~T ¢ z,L v ,
Likewise,
for the last
136
-aXo+ (T) so
(T) ~ ~(Xo-E)
and hence
= (t-C'T) ~-dE
ordp,z(T) ~ a .
Combining lemmas 5 and 6, we have
Lemma 7.
With
, ,p
and
p'
as in Fig. 2.,
i)
ai(Vy, p) + ar_i(Vz,p) 2 d
ii)
ai(Vz, p) 2 ai(Vy,p')
iii) ai(Vz, p) = ai(Vy,p')
; and for more than one value of
two or more independent sections of
Vy
vanishing only at
i
only if there are
p
and
p' .
(In fact, we conclude from Lemmas 5 and 6 that
ai(Vz, p) > ap(i)(Vy,p')
permutation
ai(Vz, p) > ai(Vy,p')
p
of
{0,...,r} , and hence that
for some
; and simi-
larly for parts (i) and (iii)).
Part (i) of (4) follows immediately from (7)(il), applied to Z = Y£+I ' p = P£+I
and
p' = p£ .
Y = Y£ ,
Part (ii) of (4)~ and thereby Theorem (3), will
follow similarly from 7(iii) once we establish
Lenuna 8.
If
£ = cm , there is at most one section
~ E Vy£
non-zero on
Y£- {P£,P~+I }. Proof:
Label the components of
zz.
¥
X0
between
Y = Y~
Zk- ~
and
E = Em
Z-k.
as in Fig. 3:
137
Suppose there are two independent sections P~+I "
vanishing only at
The pencil they span will be totally ramified at
unramified elsewhere;
in particular,
vanishing to orders exactly k
o,T ~ V
O
and
p£
there will exist sections 1
at
ql "
and
P£+I
p£
and
and
O°,T: ¢ V
Applying (7)(i) once and (7)(ii)
times, then, we have
a0(Vy, ql) = 0 , al(Vy, ql) = 1
=> ar(VZl,q I) = d, ar_l(VZl,q I) = d-i
=> ar(Vz2,q 2) = d, ar_l(VZ2,q2) = d-i
=> ar(VZk,q k) = d, ar_l(VZk,q k) = d-I
=> a r(VE,qk+l)
= d, ar=l(VE,qk+l)
But this is absurd; a pencil of degree
d
= d-l .
on an elliptic curve can't have
d-I
base points. References
[E-H]:
D. Eisenbud and J. Harris: Divisors on general curves and cuspidal rational curves, to appear. Invent. Math.
[G-H]:
P. Grifflths and J. Harris: On the variety of special linear systems on an algebraic curve. Duke Math J. 47(1980) p. 233-272.
[K]:
G. Kempf: Schubert methods with an application to algebraic curves, Publ. Math. CentrLnn, Amsterdam, 1971.
[K-L I] :
[K-e II]:
[w]:
S. Kleiman and D. Laksov: On the existence of special divisors. J. Math 94(1972) p. 431-436.
Amer.
S. Kleiman and D. Laksov: Another proof of the existence of special divisors. Acta Math. 132(1974) p. 163-176.
G. Winters: On the existence of certain families of curves. 96(1972) p. 215-228.
Amer. J. Math
PROPERTIES OF ARAKELOV'S
BY
I.
INTERSECTION PRODUCT
GERD
F A L T I N G S
Introduction In
Izv.
Akad.
duced
an
metic
surface. More
Nauk.
intersection
situation:
K
n:X
38
(1974),
product
on
the
precisely,
he
deals
is a n u m b e r - f i e l d ,
family
D=Df÷D
is a f o r m a l
sum
places
of
finite
place
divisors
A
with its
K
of
, where D~
).
curves.
F
v."
Df
= v~SE
(f) be
=
For
as
for
"the
fibre
intro-
an
arith-
following of
integers,
divisor
(f)~
=
~ v~S
r
I X(~v)
log
llfl%
field
at
v
F
v
of
f
on
X
,
with = _
where: = local
v
= ]R o r
C
.
on
on
X
X
,
S~=
infinite
of
X
function
follows:
= usual
v
ring
, (r v ~ ,
a meromorphic
(f)f
K
the
divisor
(f)f+ (f)~
defined
rv
of
Arakelov-divisor
is u s u a l
rv Fv
stands
v
An
a divisor
can
Arakelov
÷ Spec(A)
a semistable is a s u m
SSSR
O#f~
at
D
the
K(X)
in,
139
X(%
) = Riemann-surface
~fl
, KV=
I
g
2~i
k=1
~I '" "''~g
°ak ^ ~k
(X)
Then
, for
section
This
such
left
and
E
is given
by
~
G(P,Q)
with
over
p,Q ~ X ( K v )
of
are
to be m o d i f i e d ) .
can be d e f i n e d (f)
is per-
D
and
, and
for
G(P,Q)
,
the p o i n t s
is a " G r e e n ' s - f u n c t i o n "
measures
v e S
defined on
is
of
X(K)
of
the
K) usual inter-
in the points
by
X(Kv).
inter-
,
all places
E
the
and w h a t
sections
V
on the
how to d e f i n e
a divisor,
(sum over
place
v
has
intersection-product
= vZ < ~ ' E > v
v = - log where
~
a~^5 ' ,
x(~v)
the d e { i n i t i o n
are two d i f f e r e n t
section-multiplicity
f
2~i
it is clear
is the
for a finite
fibre of
=
that a d i v i s o r
of a fibre of
D
where
g=O
to any other:
essentially where
<~,~'>
F(X(K v) , ~ )
an i n t e r s e c t i o n - p r o d u c t
Arakelov-divisors, pendicular
'
an ON-bas;& of
(scalar p r o d u c t
g = genus
X (~v=~)
2 fJ
with
by
, Kv=
IIfllv =
dH =
defined
v
of the
is given
D,E G(P,Q)
by
, and can be
140
used
to d e f i n e
poles
in
P)
=
In t h i s any
Ill II2 way
arithmetic
as
on
a hermitian
above
metric
can
Arakelov's
define
the
that
theorem.
They
chapter,
the
one
2.
Summar~ We
D
on
For
Y
of
, we
I(V)
V
with have
. For
the
be
for
,
an
D = Df+D O(Df)
with
theorem
is t r u e
this
we
need
in t h e
next
for
stated sketches
of
the
for a
proofs.
V
on
Y
a hermitian V
we
. For
metric
each on
divisor
~(D)
define
.
I(V)
a divisor
is n o t h i n g D
on
Y
= I(H°(Y,~(D)))
else
, we ~
than
a volqme-
let.
I(H I ( Y , & ( D ) ) ) -I
I: are
~0R~(Y,~(D))) any
and
contains
defined
Adim(v) metric
There
a)
will
O(D)
on
places.
a Riemann-surface
10R~Y,@(D)))
Theorem
, and
just
Hodge-index
vector-space
=
A hermitian on
with
results
start
a complex
form
last
Y
is
infinite
product,
Riemann-Roch and
, which
the
intersection
metric
an Arakelov-divisor O(D)
at
show
a hermitian
a Riemann-surface
a line-bundle
We
(functions
.
can
D
X
(Q)
we
divisor
defines
O(P)
metric on
by
G(P,Q)
for
a hermitian
unique
, such
isometry ~ G R F ) 's
(up to
scalar
factors)
metrics
on
the
that
between
the
~(D)'s
induces
an
isometry
on
141
b)
If
E=D-P
, then
O ÷ H°(Y,~(E)) is
sequence
~ H°(Y,~(D)))
÷ O(D) [P] ~ H I ( y , ~ ( E ) )
Remark:
= fibre
D. Q u i l l e n
torsion",
which
If n o w
of
~(D)
can
at
define
hopefully
P . This
another
will
X/Spec(A)
an A r a k e l o v - d i v i s o r
on
x
volume
generalize
is a g a i n
has
a hermitian
via
our
volume
an arithmetic
, we have
defined
m r (X,~(D)) ~
Z m
--
M ~
for a n y A - m o d u l e ]R,X(M) ~(M)
Theorem
2:
~(D)
M
is d e f i n e d
- X(HI(x,e(D))) with
a volume
,
form on
as
= - l o g Ivol(M ~ .
m/M)/#(torsion
o f M)]
(Riemann-Roch)
--
D. (D-K) 2
. constant
(K=canonical
and
a volume-form
v~as ~r (x c~ v) ,~(D) ) ,
= X(H°(X,O(D)))
.
surface,
and we define X(D)
metric)
"analytic
on
where
÷ HI(y,~(D)) +
"volume-exact"
(~(D) [P]
D
the
class)
142
Theorem The
3:
(Hodge
intersection
the n u m b e r
of
index-theorem) f o r m has
-
-signs
signature
We can
v~ ( ~ c O m p ° n e n t s
furthermore
volumes:
Theorem
For
of
compare let
Y
group
Jacx(K))
Fv-1)
our v o l u m e w i t h mor g e o m e t r i c be a R i e m a n n
surface.
4:
For any D
this
where
is
(rank of the M o r d e l l - W e i l
#
+,-,-,-,...,-,
~ >0
of degree
there
exists
d ~ dO
Theorem If
D
D.F
> O
X
dO
, s u c h that
for a d i v i s o r
,
vol({f~F(Y,~(D)))
For a r i t h m e t i c
a
I / IIfll2 d~ Y
, this
leads
< I}) > e
-ed 2
to
5: is a n A r a k e l o v - d i v i s o r (F=fibr~,
then
As a c o r o l l a r y , relative ~x-D ~ 0
dualizing
n.D
sheaf
of bad places,
~
2
D2 > O
is e f f e c t i v e
we o b t a i n ~X
for any e f f e c t i v e
If we c o u l d b o u n d
with
that
divisor
in terms
we c o u l d d e r i v e
of
for
for
satisfies
and n>>O
g > 1 2 ~X
~ 0
.
the , and
D . K,g
, and the set
the M o r d e l l - c o n j e c t u r e
143
3. S k e t c h e s Theorem
of p r o o f s
I:
Property property denotes
b)
a) we can reduce to the d i v i s o r s
ly a big
subfamily),
Divg_1
~ Jac
into the J a c o b i a n The which
t e l l s us h o w to d e f i n e
of
divisor).
we have
a natural
Y
(or m o r e p r e c i s e -
map
of
form a l i n e - b u n d l e ~(-
We h a v e d e f i n e d
This
is done by c a l c u l a t i n g
0 )
on
a metric
Jac(Y)
on
DiVg_1
(8 = t h e t a -
on this bundle,
it is the p u l l b a c k
of a m e t r i c
its c u r v a t u r e ,
,
which
on
and we ~(-0)
indeed
comes
Jac(Y)
Theorem Just
2:
f o l l o w the u s u a l proof
Theorem
D2
on
Divg_1
Y.
to s h o w that
If
g-1
If
For
(Y)
want
from
deg(D)=g-1
of d e g r e e
I ~ R ~ ( Y , ~ ( D ) ) ) 's
is the p u l l b a c k
the volume.
F
of
R.R..
3: denotes
a fibre of
. It is s t a n d a r d
components
that
of f i b r e s of
perpendicular
to all
~ ,we this
~ , and
show that is true
if
D°F=O D
consists
so we m a y a s s u m e
such c o m p o n e n t s l s o
that
implies
that
of D
~(D) ~ P i c ° ( X )
is .
144 After
replacing
a divisor
E
D
with
H°(@(E+nD)) ~A H°(~(E+nD)) torsion.
by an i n t e g r a l (E.F)
and
I aRF(X,~(E+nD))
on
~(-@)
, this
This m e a n s
=
~,
As this
number
we can
find
that
HI(~(E+nD))
vectorspace is just a number. metric
, such
K = O for n ~ Z
vanishes,
Thus
= g-1
multiple,
that
consists
only
and the volume comes
is b o u n d e d
from
of
on this
a hermitian
below.
Thus
~((E+nD))=-log(vol(l~RF(X,~(E+n~)))-~ is b o u n d e d
above.
it can be d e r i v e d negative
By R i e m a n n - R o c h
(~torsion D2< O . More
that up to a factor
of the N e r o n - T a t e
height
of
D2 D
of H I ( x , ~ ( E + n D ) ) ) precisely,
is given
by the
in the M o r d e l l - W e i l -
group.
Theorem If
4:
voi D
denotes I
V°iD
Green's
left
< constant --
It can be shown constant
the
oe ~d2
that . FOr
function
the
side,
/ Xd
H iCj
G ( P i , P j) .d~(P 1) ...d~(P d)
integrand
this we use
for the
we get an e s t i m a t e
is b o u n d e d
that
~-operator.
by
log G(P,Q)
is a
145
Th e O r e m
5:
Apply Minkowski's
theorem
r (x, e'(~,.~))
to
to o b t a i n
an
element
o # f e r (x,~(nD)) with
I
X for
IIfit
also
/
x ( ~ v) nD+ (f)
logL1fll
Dr.
d~ < O -
are o b t a i n e d
Gesamthochschule
Fachbereich
7 - Mathematik
20
3600 Wuppertal W-Germany
just as for a l g e b r a i c
G. F a l t i n g s
UniversitMt
GauBstr.
•
> O.
The c o r o l l a r i e s
Prof.
d~ < I
v e S,
and h e n c e
Then
2
(~v)
I
-
Wuppertal
surfaces.
ON N O D A L
The c e l e b r a t e d curves
remains
conference lem
[3].
lems,
Severi
a major
Arbarello Our m a i n
some
of w h i c h
William
Fulton*
problem
on the v a r i e t y
open q u e s t i o n
has d i s c u s s e d
of d i m e n s i o n be the
have
solved.
curves
of g i v e n
singularities.
outlined
by Severi
Maltsiniotis
sion
d
is a locally
The
(imbedded)
corresponding
to a curve
of d e g r e e
(3) V(n,e)
.
If
when
d
C
which
d > e
, then
The b r a n c h e s
amon C the
tangent
n
are n o n - s i n g u l a r
appear
to other
prob-
, let
with p r e c i s e l y facts
or N o b i l e
space
d
~N
V(n,d)
nodes
and
can be p r o v e d
by m e t h o d s
[18],
-
Alibert
[13]:
submanifold
of
~N
of c o d i m e n -
.
(2)
curves
on this p r o b -
0 ~ d ~ n(n-l)/2 n
closed
At this
form a p r o j e c t i v e
[16], cf. V a n der W a e r d e n [17],
progress
its r e l a t i o n
n
The f o l l o w i n g
[2], T a n n e n b a u m
V(n,d)
degree
of d e g r e e
geometry.
nodal
problem
For any
curves
no other
(1)
Severi's
N = n(n+3)/2
set of p l a n e
some recent
is to discuss
been
of i r r e d u c i b l e
in a l g e b r a i c
purpose
§i.
The p l a n e
CURVES
of
of
one d e f o r m s
C
to
is c a n o n i c a l l y pass
through
V(n,d) V(n,e)
and c o r r e s p o n d
nodes
space
.
from
* Research partially supported N a t i o n a l Science F o u n d a t i o n .
isomorphic
is c o n t a i n e d through
The other
at a p o i n t
the nodes
to c h o i c e s
[C]
V(n,d)
d-e
of
e
to the C
.
[C]
in
"assigned"
"virtual"
of
V(n,d) points
nodes dis-
to the c o r r e s p o n d i n g
by the
space of
in the c l o s u r e
a point of
[C]
Sloan F o u n d a t i o n
branch
of
and the
147
,(n,e) (4) zurves
If C'
[C]
c V(n,d)
with
irreducible
[C']
components,
at
[C]
form the f a m i l i a r ~N
containing
for
For
normal
e
k
in
with
is the n u m b e r
of
of
C
V(n,e)
, the have
of c o n n e c t e d
compo-
picture
V(n,e)
of c o o r d i n a t e
(N-d)- plane.
n
for
of
, which
subspaces
The e s s e n t i a l
C
put
follows
- Zariski
along
of
point
independent
from R i e m a n n
connectedness
in a d e f o r m a t i o n
e < d
con- Roch.
theorem
to
the d e f o r m a t i o n s
of
nodes.
the e x i s t e n c e
d
(4), by a s s i g n i n g
nodes
branch
the v a r i e t i e s
of degree
e
C .
crossing
of the curves
"assigned"
n
S
the Enriques
In p a r t i c u l a r , degree
is a set of
is the fact that the nodes
(4) one applies
the
of
e V(n,d)
on the curves
the bl o w - u p s
where
a given coordinate
(i) - (3)
ditions
S
in the c o r r e s p o n d i n g
nents of the c o m p l e m e n t
Locally
, and
nodes, d
of i r r e d u c i b l e
for any
nodes
d ~
appropriately
nodal
(n-l) (n-2)/2 to
n
curves
, follows
lines
of from
in g e n e r a l
position.
For p l a n e the f o l l o w i n g represented nodes
quartics, diagram.
the p o s s i b l e Each
irreducible
by a c o r r e s p o n d i n g
indicate
V(4,6)
deformations
nodal
nodal
from
component
curve.
V(4,d)
V(4,5)
curves
are i n d i c a t e d of each
The c i r c l e d
into
in
V(4,d)
"virtual"
V(4,d-l)
V(4,4)a
is
148
V(4,3)d
V(4,4)b
V(4,3)c,e
< V(4,2)
Let problem
V(n,d)*
= {[C]
is to show that
analyzing
the v a r i e t y
combination to showing
with
irreducible
In fact,
, one
however,
d
in g e n e r a l
of degree
of
it w o u l d n
Severi
(irreducible).
of lines
of any c o m p o n e n t on
The
sees that this p r o b l e m V(n,d)*
suffice
By
position,
in
is e a u i v a l e n t m u s t meet
to show that any
can be d e g e n e r a t e d
g
no one has p r o v e d
degeneration, plane curves.
as follows. --> M
is i r r e d u c i b l e }
to a nodal
nodes.
by all other
: V(n,d)*
is c o n n e c t e d
V(n,n(n-l)/2)
curve
any n o n - t r i v i a l
underscored
V(n,d)*
By i n d u c t i o n
nodal
curve w i t h more
shared
I C
(I) - (4)
.
V(4,0)
c V(n,d)
that the closure
V(n,n(n+l)/2)
have
V(4,1)
Let
g -
be the c a n o n i c a l
i.e.
that p l a n e
nodal
a degeneration
The e x t e n t (n-l) (n-2)/2
curves must
which
is not
of our i g n o r a n c e - d
m a y be
, and let
m a p to the m o d u l i
space
M
g
of
149
n o n - s i n g u l a r curves of genus
Challenge. that M
T(V)
If
V
g .
is an i r r e d u c i b l e c o m p o n e n t of
is not c o n t a i n e d in a complete
V(n,d)*
, show
(compact) s u b v a r i e t y of
g Although
Mg
can contain c o m p l e t e subvarieties,
one does not
know how large their d i m e n s i o n can be.
§2.
I r r e d u c i b i l i t y of
One of Severi's reasons for proving give a proof of the i r r e d u c i b i l i t y of geometry.
M
Mg
V(n,d)*
using m e t h o d s of a l g e b r a i c
g
Indeed, for large
n , Y
versely,
the i r r e d u c i b i l i t y of
M
V(n,d)*
is i r r e d u c i b l e for s u f f i c i e n t l y large
D e l i g n e and M u m f o r d Mg
g
maps
i r r e d u c i b l e was to
V(n,d)*
onto
M
(Con-
g
has been used in proofs that n .)
[5] c o n s t r u c t e d a c o m p a c t i f i c a t i o n
Mg
of
, w h o s e b o u n d a r y points c o r r e s p o n d to stable curves whose irredu-
cible c o m p o n e n t s have g e o m e t r i c genus the i r r e d u c i b i l i t y of
Mg
n o n - s i n g u l a r curve of genus
less than
g .
They show that
is e q u i v a l e n t to the a s s e r t i o n that any g
can be d e g e n e r a t e d to a stable curve.
Until r e c e n t l y even this simple c o n s e q u e n c e of Severi's a s s e r t i o n had no proof w i t h i n a l g e b r a i c g e o m e t r y - although of course there are w e l l - k n o w n t o p o l o g i c a l and analytic proofs of the i r r e d u c i b i l i t y of M
g Thanks to a b e a u t i f u l c o n s t r u c t i o n of Harris and M u m f o r d
one can now give a simple proof that such d e g e n e r a t i o n s In place of the space of node curves, space
Hn, b
of n - s h e e t e d coverings of
[i0],
always exist.
one m a y c o n s i d e r the Hurwitz ~ i , with
b = 2g+ 2n- 2
simple b r a n c h points, w h i c h is an etale c o v e r i n g space of the space of b - tuples of points in
~I
(modulo a u t o m o r p h i s m s of
and M u m f o r d construct a c o m p a c t i f i c a t i o n of sist of s t a b l e curves
Hn, b
~ i ).
Harris
whose points con-
C , t o g e t h e r w i t h b r a n c h e d coverings
C --> D ,
150
as above, which
where
each
ramification
component
may occur,
of
D
It follows
that each c o m p o n e n t
C
For details
we refer
to
It is t a n t a l i z i n g p rove
V(n,d)*
through branched
that
of
a clever
of
n
Zariski abelian
and P r i l l
fundamental ed
pl
meeting
formula the proof.
problem
n
lines
lets the b r a n c h
points
to
of a
at a con-
One may hope
V(n,d)*
may
similarly
and m u l t i p l e planes.
that the f u n d a m e n t a l
nodal curve
C
, from
in general
assertion
in [7] and [i] and
classical
in g e n e r a l [15] were
results
[91 .
Severi's
was p r o v e d by D e l i g n e new
proof
~I(P 2 -C)
is
assertion
that
Special
assertion,
cases
C
of this
by A b h y a n k a r
for the a l g e b r a i c [4].
What was n e e d -
"connectedness
of Bertini;
Another
group
position.
Zariski's
and
to
attempt
argument.
[15].
papers
Severi's
a curve
of the spaces
to S e v e r i ' s
generalize
concludes
at a point.
appealing
of
- Hurwitz
one also arrives
without
group
the o r i g i n a l
of
to lines
beyond m e t h o d s
which
come together,
Zariski's
[9] d e d u c e d
were proved,
which
this proof with
if one n a i v e l y
Severi's
can be d e g e n e r a t e d
[i]
~i
for any plane
from the R i e m a n n
by d e g e n e r a t i n g
compactification
§3.
over
one of these r a m i f i c a t i o n
be rational,
to c o m p a r e
copies
be used to rescue
three points
[8].
In fact,
covering
figuration
must
irreducible
a point.
at most
and at least
points must be simple. of
has
theorems",
for this we refer
of Nori
[14]
to
is d i s c u s s e d
in §4.
When equation that
C
is an i r r e d u c i b l e
f(x,y)
~ i ( ~ 2 - C)
ramification For integer
= 0 , the a s s e r t i o n
that
is cyclic
n
curve
any plane
of order
in the surface curve
k ~ 2 , let
ing c o r r e s p o n d i n g
node curve
C
Yk -->
with p2
to the e q u a t i o n
of d e g r e e
n
~ i ( ~ 2 - C)
is a b e l i a n m e a n s
, or that
z n = f(x,y) equation
z
k
is simply
= f(x,y)
by an
of the
the c o m p l e m e n t
f(x,y)
be the finite
defined
connected.
= 0 , and any
cyclic
k-sheeted
;
is taken
Yk
coverto be
151
normal,
and the r a m i f i c a t i o n
at infinity.
Let
Zariski
[20]
invariant
of
families
Zk --> Yk
C .
of the
allowed
numbers
Hodge
As Zariski mately
related
irreducible ~ I ( P 2 -C) k
•
cyclic.
is abelian,
If
out,
the h y p o t h e s e s Uk
[20]
is the c o m p l e m e n t
~l(Uk)
is also cyclic,
Zl(Uk)
~l(Zk)
takes
Lazarsfeld
the n o r m a l v a r i e t y g roup
Yk
and first betti
divisors
p2 _ C . L
in the r e s o l u t i o n
these betti
Zk
is inti-
Suppose
C
transversally.
that
~l(Zk)
inverse
is If
= 0
for all
is i n f i n i t e
image
of
C0L
in
to zero.)
is simply connected. of
by any
respectively.
of
out that for any curve
number
of equiva-
and the s u r j e c t i o n
a generator
has p o i n t e d
studied
~ i ( ~ 2 - (CUL))
of the
Yk
could d i s t i n g u i s h
be r e a l i z a b l e
number
of
showed
of
as a subtle
polynomials,
imply that
Z k , then -->
have
the line at infinity Zariski
Zk
number
cannot
the first betti group
the line
to show that certain
[12]
and A l e x a n d e r
to the f u n d a m e n t a l
and meets
(Indeed,
equations
of
the same number
idea
[6] and L i b g o b e r
theory
points
this
over
of s i n g u l a r i t i e s
number
same degree with
by P l ~ c k e r ' s
Esnault
and perhaps
that this b e t t i
He also used
p lane curve•
C
be a r e s o l u t i o n
He showed
lent singularities.
using
over
used the first betti
of curves
invariants
occurs
Zk
all
C
Thus
k
,
the f u n d a m e n t a l
"come from"
of s i n g u l a r i t i e s
, and any
the e x c e p t i o n a l
Zk --> Yk
"
In fact one
has the following:
Theorem
(Lazarsfeld).
variety,
f : X -->
tive d i m e n s i o n a l restriction ~I(X)
Let
pn
be a normal,
a finite morphism.
locally
f-l(v)
X
closed
--> V
is
projective Suppose
subvariety
V
of
(set-theoretically)
n-dimensional
there ~n
is a posi-
such that the
one-to-one.
Then
= 0 . Proof.
taking
One may r e d u c e
generic
tive m o d e l
of
to the case w h e r e
hyperplane
sections.
V
: C -->
, and
g
~n
Let
C
V
is a curve,
be a n o n - s i n g u l a r
a morphism
that maps
C
by projec-
"
152
b i r a t i o n a l l y onto the closure of
V
in
pn
Let
The h y p o t h e s i s implies that the p r o j e c t i o n from and g e n e r i c a l l y one-to-one; h o m e o m o r p h i c a l l y onto
since
C
C' =
C'
to
is n o n - s i n g u l a r ,
(C × C C'
nX)red
is finite maps
C .
Now apply D e l i g n e ' s v e r s i o n of the c o n n e c t e d n e s s t h e o r e m ([4],
[9]) to the product m o r p h i s m
F : C x X --> ~ n x ~ n
, which
yields the s u r j e c t i v i t y of
(F-I(A~n))
--> ~l(C × X)
i.e. of the diagonal m a p
~I(C')
~I(C')
m u s t be zero.
> ~I(C)
' ~l(X)
§4.
M. Nori
Zariski problem,
Abhyankar
Nori's theorem.
~ 2 , as sought for and p r o v e d in special cases
[i] and Prill
s e l f - i n t e r s e c t i o n number, r(D)
Theorem
[15].
of
D i.e.
on a normal surface, YCl(0(D ) 2
denote the number of nodes of
(Nori [14]).
a nodal curve on
of the s o l u t i o n to
but he also gives analogues for nodal curves
For a Cartier d i v i s o r
let
But since
He not only gives two d i f f e r e n t proofs for the
on surfaces other than by
× Zl(X)
[14] has p r o v e d vast g e n e r a l i z a t i o n s
Zariski's problem• original
--> ~I(C)
l
X .
Let
X
denote by
D2
For a nodal curve
the D ,
D .
be a n o n - s l n g u l a r p r o j e c t i v e
surface,
Assume that for each i r r e d u c i b l e c o m p o n e n t
C , D 2 > 2 r (D)
Then the kernel of the h o m o m o r p h i s m
~l(X- C) --> Vl(X)
C D
153
is a b e l i a n .
In p a r t i c u l a r , abelian.
Note
and
~
r(D)
theorem.
noted
by Abhyankar,
(Nori
cible
projective
(a)
Let a point
of
shows
on
2
and
on
algebraic
from Nori's
proof,
as w e l l
resist
fundamental
is f o l l o w i n g
is
D2 = m2
follows
we cannot
C)
pre-
group.
lemma,
as
As
which
is
of
be a f i n i t e non-singular, the
f(A) u C
morphism
of irredu-
and
C c X
following
at a p o i n t
.
An
satisfy
Y
let
conditions
of
function
is G a l o i s ,
V
of
Y
b_ee
o n an
f(A)
is a n o d e .
I(V)
=
h o g
I ~(V)
satisfying
X
intersect.
in t h e
, so w e m a y
c G(P)
= V}
group
G
field
of
assume .
For
Y any
,
[ O l V = id v}
(a) a n d
local
is c y c l i c ,
:
of
with Galois
{~ c G(V)
of
(b) m u s t
, set
= {~ e G
I(A)
field
X
analysis
(a) a n d
b y its n o r m a l i z a t i o n
G(V)
be a c u r v e
into
X
which
of t h e
f : Y -->
that
X
:
I(A)
f
However,
Consider
replace
(i) Factor
.
, then
the g e n e r a l
point
with
Y
Y
subvariety
A
assertion
~I(X-
> 2 r(f (A))
One may
A
X
for t h e
curve.
A
extension
irreducible
m = deg(D)
for
: Y -->
surfaces,
any two curves
is n o r m a l
f
singularity
f(A)
a Galois
[9])
Let
curve
Proof.
proof
branch
Any
(b)
paper
the essential
[14]).
(reduced)
irreducible
• and
then
interest.
Lemma
Then
to Nori's
beautiful
connected,
, so Z a r i s k i ' s
to non-compact
of i n d e p e n d e n t
the
is s i m p l y X = ~2
(m-l) (m-2)/2
We refer
his
X
that when
generalizations senting
if
(b), a n d
fundamental
G(P)
.
groups
is a b e l i a n ,
c G(A)
let
and
P
be
([i],
[7],
154 g Y-->
Let
B = g(A)
By g e n e r a l
, D = h(B)
(2)
A = g
(3)
h
(4)
B
In fact,
from
h
(3) a n d
if
(5) it f o l l o w s
B of
mal bundles
ramification
(B)
theory,
,
onto
D
ramification
locus
of
at e a c h p o i n t
,
h
.
and
D
are
Y/G(A) agree
and
by
of
B
.
that
= D 2 - 2 r(D)
the non-singular
(6) are t h e d e g r e e s
B -->
.
birationally
B 2 - 2 r(B)
two sides sions
in t h e
is 4 t a l e
(6)
Indeed,
B
-i
X
(i), o n e h a s
(5)
From
maps
is n o t
h -->
Y/G(A)
5 -->
models
of t h e n o r m a l X
; and
of
bundles
B = D
by
and
B
of t h e
(3), w h i l e
D
, the
immerthe nor-
(5).
In p a r t i c u l a r , (g'B) 2 = B 2 ~ D 2 - 2 r ( D )
Note
that,
(a) a n d and
A2
Cartier
by
(2),
g*B
(b) s u p p o r t s were
a Cartier
two such
divisors
A1
is s u p p o r t e d
~21 > 0 , A which back
contradicts
the
to a r e s o l u t i o n
theorem
divisor
curves which and
Hodge
A2
A A
were
would
index
surfaces;
. with
Thus
any
~2
> 0
disjoint,
the
A
satisfying But
if
A1
corresponding
satisfy
A2.A 1 = 0 ,
A~22 > 0 ,
theorem.
of s i n g u l a r i t i e s
on non-singular
on
> 0
of
or cf.
(By p u l l i n g Y
, one may
[ii]).
D
the divisors
use
the
index
155
To prove the t h e o r e m for the algebraic f u n d a m e n t a l group, f : Y ~>
X
be a Galois covering,
Galois group
G
g e n e r a t e d by the groups
i r r e d u c i b l e components of G , and
Y/H --> X
abelian. I(A I)
f-l(c)
I(A 2)
cyclic groups
A1
commute by
I(A)
Then
is unramified.
If two c o m p o n e n t s
and
and let
I(A) H
be the subgroup of the , as
A
runs over the
is a normal subgroup of
It suffices to show that
and
(i).
H
A2
let
intersect at
H
is
P , then
The lemma then shows that all the
must commute with each other, w h i c h makes
H
abelian.
Remark.
In c h a r a c t e r i s t i c
p , the lerm~a and its proof are v a l i d pro-
v i d e d the r a m i f i c a t i o n is tame. d a m e n t a l group.
The t h e o r e m is true for the tame fun-
L a z a r s f e l d ' s theorem,
as proved in §3, also holds for
the a l g e b r a i c f u n d a m e n t a l g r o u p in a r b i t r a r y c h a r a c t e r i s t i c .
References
i.
Abhyankar, S., Tame coverings and f u n d a m e n t a l groups of algebraic varieties, I,II, Amer. J. Math. 81(1959), 46-94; 82(1960), 120-178.
2.
Alibert, D. and M a l t s i n i o t i s , G., G r o u p e f o n d a m e n t a l du c o m p l ~ m e n t a i r e d'une courbe ~ points doubles ordinaires, Soc. Math. France 102(1974), 335-351.
Bull.
3.
Arborello, E. and Cornalba, M., A few remarks about the variety of irreducible plane curves of given degree and genus, preprint.
4.
Deligne, P., Le groupe f o n d a m e n t a l du c o m p l 4 m e n t d'une courbe plane n'ayant que des points doubles ordinaires est ab41ien, S4minaire Bourbaki n ° 543, Springer Lecture Notes 842(1981), i-i0.
5.
Deligne, P. and Mumford, D., The i r r e d u c i b i l i t y of the space of curves of given genus, Publ. Math. I.H.E.S. 36(1969), 75-100.
6.
Esnault, H., Fibre de M i l n o r d'une cone sur une courbe plane singuli~re, Invent. Math. 68(1982), 477-496.
7.
Fulton, W., On the f u n d a m e n t a l group of the c o m p l e m e n t of a node curve, Annals of Math. 111(1980), 407-409.
8.
Fulton, W., On the i r r e d u c i b i l i t y of the moduli Invent. Math. 67(1982), 87-88.
9.
Fulton, W. and Lazarsfeld, R., C o n n e c t i v i t y and its a p p l i c a t i o n s in algebraic geometry, Springer Lecture Notes 862(1981), 26-92.
space of curves,
"Anzi che trarre delle c o n c l u s i o n i di p r i o r i t ~ il cui interesse per la scienza ~ sempre scarso, o dar troppo peso a q u a l c h e d e b o l e z z a ed agli errori di grandi scienziati, ~ m e g l i o r i l e v a r e le nuove c o n f e r m e di quel fatto che c o n t i n u a m e n t e si p r e s e n t a n e l l a storia della matematica: che alla c o n e s c e n z a completa, generale, d e l l ' e n t e o del r i s u l t a t o esatto si ~ giunti non in un sol t r a t t o e per o p e r a di un solo, ma per op e r a a l t e r n a t a o s i m u l t a n e a di vari, p a s s a n d o per pi~ gradi si di g e n e r a l i t ~ che di rigore!" Corrado
Segre
[1892],
concluding
ABOUT
paragraph.
THE E N U M E R A T I O N
OF C O N T A C T S
by
William
FULTON,
Steven
KLEIMAN I and Robert
I. Introduction.
The e n u m e r a t i o n of v a r i e t i e s ally is large number 1864.
of c o n t a c t s
in a p - p a r a m e t e r and unobvious.
of conics It is not
touching 7776,
Preliminaries.
is a f a s c i n a t i n g
family
touching
For example,
p
subject.
that
is 65 ,
over the c o m p l e x
reaffirmed
in 1859.
This
sidered
many times
over the years
as Steiner
example
The number
given varieties
5 given oonics is 3264 just asChasles
Bishoff
1
MacPHERSON
asserted
numbers, found
increasing
the
in
in 1848 and
has been c o n s i d e r e d
with ever
usu-
and recon-
degrees
of clarity,
This report was p r e p a r e d w h i l e the author was on s a b b a t i c a l leave from the M a s s a c h u s e t t s Institute of T e c h n o l o g y and a v i s i t i n g p r o f e s s o r at the U n i v e r s i t y of Copenhagen, Denmark. It is a p l e a s u r e for him now to thank the M a t h e m a t i c s Institute of the U n i v e r s i t y of C o p e n h a g e n for its h o s p i t a l i t y and Dita A n d e r s e n for her fine typing.
157
rigor and c o m p l e t e n e s s , given
conics
(so 3264)
are
in general
conics
multiplicity of q ua d r i c s
touching
cubics 783,
touching
until
K£fiec
them,
9 given
found
after
touching
680,
position,
I, was not taken
just as Schubert orously
but the notable
1980.
then there
1974.
quadrics.
A second
This
However,
12 given quadrics. found
number
that,
when
finitely
many
and c o u n t e d step
the
with
is the number 841,
088,
it was not e s t a b l i s h e d
rig-
A third example
just as S c h u b e r t
are
each n o n d e g e n e r a t e
until
in 1870.
step of showing
is the n u m b e r
It is most
in 1874.
is 666,
likely,
However,
of t w i s t e d
5, 819,
539,
it has not been ver-
as of this writing.
in 1874. The others
first
like them has been to reduce
the numbers ear
step in the d e t e r m i n a t i o n
in the r e l a t e d
spaces -points, lines,
reduction, al case
it turns
is the m a j o r
The current and proves
etc.
is always
- in e v e r y
of the
final
number
which
generalizes
of i n d e p e n d e n t
interest.
several
itself.
of proof.
Section
in the gener-
Section
statements
Section
2 states that qual-
3 discusses
The section
the lemma of S e c t i o n
Finally,
Such a
work.
include
of v i e w and methods
a proposition,
The r e d u c t i o n
of
are lin-
combination.
with preliminaries.
which
and of
the g i v e n v a r i e t i e s
possible!
continues
the main results,
points
in w h i c h
planes,
numbers
the p r o b l e m to the d e t e r m i n a t i o n
theme of the p r e s e n t
section
ify the s i g n i f i c a n c e some other
out,
cases,
of the above
closes
2, and w h i c h
4 discusses
various
with is
open
problems.
F r o m now on, arbitrary
characteristic
Subvarieties variety), eter
the g r o u n d
will
. Varieties
be c l o s e d
T ,
but the total
X
will space
be a l g e b r a i c a l l y
will be r e d u c e d
and proper
but they may be r e d u c i b l e
family of v a r i e t i e s
variety
field will
(not empty
and of impure be p a r a m e t r i z e d
closed
algebraic
schemes.
nor the w h o l e dimension.
and of
ambient
A p-param-
by a p - d i m e n s i o n a l
need not be flat over
T .
The
158
phrase
"almost
of
whose
T
Fix The
all
X"
will mean
complement
is at m o s t
a subvariety
(projective)
set o f p a i r s
V
of the
conormal
(P,H)
such that
containing
the
in
where
is t h e
,
hyperplanes For
of
~N
embedded
in
V
hard
endowed
subvariety
of
the
variety
Pr2CV
"orthogonal"
V ,
and only
V
V*
(1.1)
dim(V)
and equality
holds
The
H
in a c e r t a i n
conormal
coincides
with
that,
V
of
if
V* ,
There
the
CV ,
Wallace
then
is r e f l e x i v e , contact
is a u s e f u l
P
(see K l e i m a n
V
is e q u a l
VH
of t h e V*
H
if
In g e n e r a l ,
It s i t s space of
as t h e
It m a y H
,
be
touches locus
P. of
ideal.
is d e f i n e d
as t h e
is a
d-plane,
for a n y
simple
and the
is
N-1 .
singular
V V
and
Pr2: C V ~ ~ N
Jacobian
of
(TpV)* ,
V
is d e f i n e d
at w h i c h
.
of t h e
the
CV ~ ~N.
to the
~N
two
then
V*
point
are e q u a l
P
if
So
> N-l-dim(V*)
if
V
CV* V
subset
also
lies
VH
of
in
be c a l l e d
V**
= V
is e q u a l
and,
namely,
clearly
V
V* .
~N x ~N
,
and
if
CV*
reflexive.
It is e v i d e n t
for a n y
point
to t h e
for r e f l e x i v e n e s s ,
[1980]);
Furthermore,
= N-I
open
will
then
,
is l i n e a r .
+ d i m ( V H)
locus
criterion
subset
closure
projection
Pr1:
of
structure
dense,
variety
locus
second
For example,
dimH(V*)
for all
H-contact
space
TpV .
dimension
VH
if and o n l y
(1.2)
space
is o f p u r e
contains
is l i n e a r .
of
CV
(N-1-d)-plane.
obviously if
~N.
point
tangent
projection
scheme
as t h e
N-space,
(or r e c i p r o c a l )
of
is d e f i n e d
projective
of the
that
projective
is a s i m p l e
of p o i n t s
to p r o v e the
H
b y an o p e n
dual
the
first
locus
with
The dual
is t h e
via
CV
P
that
H ,
parametrized
N-dimensional
(embedded)
fiber o v e r
o f as t h e
It is n o t V D H
Note
a hyperplane
scheme-theoretic
thought
~N
X
(p-1)-dimensional.
variety
a hyperplane ~Nx~N
all
linear
due
to
is r e f l e x i v e
simple space
H
TH(V)*
C . Segre
and
if a n d o n l y
if
159
the map CV).
Pr2:
C V ~ V*
is s e p a r a b l e
In c h a r a c t e r i s t i c For
i = 0,..-,N-1
(1.3)
will
ri(V)
be c a l l e d
ducible hard
zero,
the
is,
the very
i-th rank d ,
r. (V)
of
therefore,
important
= IPrl*c1(0(1))i
of dimension
to see t h a t
V
(smooth on a dense
is just
always
subset
of
reflexive.
nonnegative
integer
Pr2*c1(0(1))N-1-i[cv]
V .
to the
open
(It is e q u a l ,
(d-i)-th
class
the degree
when
of
V
V .)
of the polar
is i r r e -
It is n o t locus
of d i -
1
mension of
i ; this
simple
plane,
locus
points
then
P
VI,...,V m
of
(TpV)+A
It is e v i d e n t are
V
r. (V) l
It is v e r y
e a s y to
(1.5)
r
if
A
in
V
of t h e
is a g e n e r a l
scheme
(N-2-i)-
a hyperplane.
components
= ri(Vl)
see u s i n g
closure
is a d d i t i v e
irreducible
r.(V)l
as t h e
such that,
is e x a c t l y
that
the
(1.4)
is d e f i n e d
+'''+
in
V ;
of
V ,
is,
if
then
i(Vm ) "
the projection
formula
(V)
= 0
(V)
= deg(V)
for
= 0
i < N-l-dim(V*).
for
that
i > dim(V)
that
,
l
(1.6)
r
i = dim(V)
and
1
(1.7)
r. (V) l
It is n o t ducible,
h a r d to p r o v e
r
It is e v i d e n t
r
l
and
al c e n t r a l
Urabe
for
if
V
(V)
= r
lovely observation
(3.3),
(4.1)
(V) % 0
that,
(1.9)
and
(see H e f e z - K l e i m a n
[1983]),thatif
V
is i r r e -
then
(1.8)
(This
for
[1981].)
(4.2)) t h a t , projection,
if
N-l-dim(V*)
is r e f l e x i v e ,
N-I -i
(V*)
contains
then
for all
hard
< N-2
.
then
the essence
It is n o t dim(V)
< i < dim(V)
i.
of
(3.6),
to prove and
if
p:
Piene
(compare V ~ ~N-I
[1978]
Piene
[1978],
is a g e n e r -
160
(1.10)
r
and that,
if
H
r
if
(V)
= r i(pV)
is a g e n e r a l
(1.11)
Finally,
1
V
i
(V)
= r
is a s m o o t h
(1.12)
r
(V)
for
i = 0,..-,N-2,
hyperplane,
i-I
(VnH)
then
for
hypersurface
i = I,.-.
of d e g r e e
= m(m-1) N-1-i
for
N-I ' "
m,
then
i = 0,...,N-I
,
1
because
CV
Let
= ]P(0v(m-1)) ,
10'''''IN-1
(1.13)
will
be
r(v)
troduced
of the
indeterminates.
of
V.
condition
the term
is e a s y t o c h e c k . The expression
= r0(v) 1 0 + . . . + r N _ I (V) IN_ I
be c a l l e d the m o d u l e
the module
as
(More p r o p e r l y ,
to t o u c h
in c o n n e c t i o n
with
V,
it s h o u l d
following
arbitrary
be c a l l e d
Chasles,
conditions
who
in-
on c o n i c s
in
1864.) A second
subvariety
make
a contact
have
a common
of contact. if
H
with pair
X
V , (P,H),
to the
P
will
dim(X)
X
if b o t h X*
(TpX)
be called
and X
V and
touches
are t h e
same
Consider
V* ;
touch V
just
are
as t h o s e
(P,H)
of
if a n d o n l y
X*
will
point
CX
be c a l l e d
of
spaces
then
Note
X
and the
and of
TpX
o r to
and
CV element
V
and
TpV ,
that,
if
< N-I ,
if t h e y m e e t . X
the elements
and
varieties
V ,
(TpV),
+ dim(V)
then
a p-parameter
+
proper.
reflexive,
in fact,
s a i d to t o u c h
two c o n o r m a l
span of the tangent
(1.15)
then
be
is a s i m p l e
H =
contact
will
and then
(1.14)
then the
~N
if t h e
If, m o r e o v e r ,
is e q u a l
of
touches
Note V
of c o n t a c t
also
that,
if a n d o n l y
if
of
V
X
and
V* .
family of
subvarieties
X
of
~N.
For
161
each sequence of nonnegative Zjk = p , Jk
the number
general
number.
for
j = 0,...,N-I
characteristic
reflects
can be formed.) dependent
simultaneously
will be called the
is equal to the
The word
"elementary"
are finite and w e l l - d e f i n e d
of dual varieties immediately
then the
(JN-I
(j0,..-,JN_1)-th
,...,j0)-th characteristic
X* .
"char-
suffice to characterize reflects
These
facts and others
from the t h e o r e m below,
(that is, in-
Any subfamily
has the same characteristic
are reflexive,
or elementary
1864. The word
of the choice of the general k-planes). X
touching
are the basic ones, out of which the others
These numbers
taining almost all
such that
characteristic,
of contacts.
the fact that these numbers
V1,''',V
number,
X
the fact that these numbers
the family in enumerations
X
of
(The terms are primarily due to Chasles,
acteristic"
all
(j0,-.-,JN_I)
I(j0,...,JN_I)
k-planes
(j0,...,JN_1)-th
integers
numbers.
con-
If almost
characteristic
number
number of the family
about the numbers
applied with the k-planes
result
as
p .
2. The Main Results. Theorem.
Given a p - p a r a m e t e r
p
other subvarieties
U
of the p-fold self-product
that,
family of subvarieties
X
of
~N
and
V I , . - . , V p , there exists a nonempty open subset
for any p-tuple
of the a u t o m o r p h i s m
(gl,...,gp)
replaced by their translates
in
U ,
after
group of
~N
VI,..-,V p
giV1,.--,gpVp , the following
such have been
statements
will hold: (a) city,
(i) The number of simultaneously
X ,
touching
each
X
VI,...,V p
given by the product of the modules (2.1) The product
counted with its natural multipliis finite.
(ii) The number
is
of the V's,
r(v1)...r(v p) . is evaluated
placing the monomial
10
by formally m u l t i p l y i n g 90
.-.IN_ I
JN-I
it out and then re-
by the c o r r e s p o n d i n g
characteris-
162
tic
number
for some
I(j0,...,JN_I ) i
and a l m o s t
all
(2.2)
dim(X)
(2.3)
dim(X*)
(b)
The e x p r e s s i o n
is s e l f - d u a l ; the
that
corresponding
when
almost
all
(c)
If an
X
is p r o p e r , Vi
lies
with in any
and
any g i v e n X
have
almost
that
contains
X
not
2 .
(d)
Suppose
all T
X
and
that
appear
X
CV, i
m,
and a l m o s t
all
family
(that
and
VI,...,V p
with
the
pears
in the
count
power
of the
characteristic
ous
contacts
of
X
with
X
if,
p
(iv)
more
and the
the
of
X
and
(Pi,Hi)
itself
belongs
for no
(Qi,Li)
is,
contacts
continuously
i
lies to does
distinct
generally,
from
when
characteristic
and t h a t
rational
m) ; however, set
map
m = 0
is
almost from
if a l m o s t
appears in the count in (a) with
irreducible,
is the
VI,-..,V p .
with
VI*,...,Vp*,
0 . When the c h a r a c t e r i s t i c are
c
terms,
(Pi,Hi)
is i r r e d u c i b l e
same m u l t i p l i c i t y and
X ,
nor,
T
when characteristic is X
0 ,
is of degree
of the
X
element
space
appear infinitely often. Then each
multiplicity
all
reflexive
in the
0
VI,-..,Vp
touching
and s i m i l a r l y ,
VI,...,V p
times
of the
varies
(iii)
with
are
touching
X*
that
is
into the Hilbert scheme of ~ N
all
of
characteristic
the p a r a m e t e r
m
CX
almost
Vi
X
of c o n t a c t
outset,
of
of
(i) e a c h
at the
subfamily
all
then
given
with
is
reflexive.
of
subset
the
are
subset
open
when
number
the e l e m e n t
dense,
,
number
open
contact
number
or
up to the o r d e r
for the
i
The
<_ N-2 .
for the
VI,...,V p
for e a c h
a second
(Pi,Hi)
+ dim(Vi*)
VI,...,V p ,
dense,
(iii)
either
it c o i n c i d e s ,
and
was
family.
+ d i m ( V i) _< N-2
expression
that
given
X ,
(a,ii)
touches
in any
X
in is,
X
(ii)
of the
then
mcq ,
number
is p o s i t i v e
each
where
of d i s t i n c t
X q
apis a
simultane-
163
Remark.
In
when
characteristic
the
(d), u s u a l l y
of t h e m o d u l e s . are reflexive
Corollary. number
of
either 2 ,
~N
smooth
subvarieties
does
Moreover,
and the
In
a smooth
complete
hypersurface
of degree
Proof
corollary.
that
there
IV,
fore the corollary Proof
of the
The t h e o r e m
when
almost
at
least
intersection
obviously
given X
and
VI,..-,V p of
(c,iv).
a n d at m o s t
a finite
touching if,
p = (NNm)-
for e a c h
a smooth
of degree
q = 1
by the product
by virtue
one
m
> 2 ,
all 2
position
of d e g r e e
In fact,
number
is n o t
in g e n e r a l
i ,
surface
~ 2 ,
I
Vi
is
of degree
or a s m o o t h
> 2 . The hypotheses
from
(plus
on
passing
(a,i),
imply
of t h e
hand,
and
of the
(see
it is e v i d e n t p
points.
(e,III)
a remark
geometry
r 0 ( v i) % 0
through
(a,ii)
2 parentheses,
is a r e f l e c t i o n
Vi
On t h e o t h e r
I hypersurface
follows
theorem
the
of degree
D, p p . 3 5 9 - 6 5 ) .
is e x a c t l y
c = I .
divide
is a l w a y s
hypersurfaces
VI,-..,V p
[1977],
not
c = I
there
a curve
Kleiman
and
characteristic
a point,
of the
q = 1
and
of the
ThereTheorem.
an e x a m p l e ) .
incidence
correspond-
ence,
I = { (P,H)
(2.4)
Note
that
I
automorphism that
I
V
Recall
that,
only
is a h o m o g e n e o u s group
contains
variety
if
of
CX
6 m N x~NIP
~N.
A u t ( ~ N) . CV ,
which
Note
CV .
Note
gCV
a second
These
under that
.
the I
induced
= CgV
N-I ,
for all
subvariety
observations
action
of t h e
is of d i m e n s i o n
is o f d i m e n s i o n
that
by definition, meets
space
6 H}
X
2N-I
for e v e r y
g
in
and
sub-
A u t ( ~ N) .
touches
V
form the backbone
if a n d of t h e
proof.
(Remark. derivation
One theme
of the proof
of B e r t i n i ' s
one
in H o d g e - P e d o e
and
it is w o r t h
theorem.
([1952]
considering
Thm.
is i l l u s t r a t e d
in t h e
This derivation I, p.153)
in p a s s i n g .
and
In t h e
following
is a v e r s i o n
in K l e i m a n setup
of the
([1974]
above,
short
(I0~,
by dimen-
164
sional
transversality,
A u t ( ~ N) hence,
such that
the
sally
smooth
This
CX
Proceeding
loci of
of
U (cx)XP
where
viously,
Fp
p+1 X
factors varies
(7.2)), after
lates,
the set
(2.5)
the
intersectands
0 ;
in p o s i t i v e
of
X
point of
of
X
of
for all
g
in
which
open
of
with,
Vi
be any c o m p a c t i f i c a t i o n form the closure
~N ,
parametrized
of
X
has diof
T O . In
Fp
of the union
by
TO .
I ~p
of
Ob-
p
is transitive,
[1974]
and V a i n s e n c h e ~
Aut(~N~
(see K l e i m a n
by general
of e a c h
transversally
each point q
in
Sp
Sp
lies
Furthermore, in c h a r a c t e r i s t i c
appears
w i t h the same
of the c h a r a c t e r i s t i c .
open
represents for each
Moreover,
"intersectand".
(differentiably)
trans-
D Fp
subsets an
i ,
X
above,
in the s u b f a m i l y
an e l e m e n t
it may be a s s u m e d
and that
Hi
it may be assumed
of contact
that
Pi
is a simple p o i n t
with
V. l
M =
(Tp.X) 1
is proper.
+
(Tp V i) 1
Indeed,
parametrized (Pi,Hi)
is a simple of
V.* l
(2.6)
of
be any open
complement
is the i-th projection.
subset
Moreover,
and of
TO
have been r e p l a c e d
is a power
Sp
let
(Pr1-1CVl)N...D(prp-lCVp)
appropriate
Vi .
The contact
U ;
transver-
for s u b v a r i e t i e s
family w h o s e
subfamily
on
characteristic,
together with
U
(differentiably)
theorem
xT ,
results
pr i
meet
By using the
TO
meet
subset
pN .
Vl,.-.,V p
where
in any given dense,
that each point
gV
T xp
in the
action
Sp =
by
I
is of d i m e n s i o n
finite,
gCV = CgV
of the theorem,
and let
transversality
multiplicity,
and
space of the
p-1 ,
[1978]
becomes
X
w i t h the p r o o f
Since the natural by general
not meet
open
in any characteristic.)
at most
the p r o d u c t
a nonempty
is Bertini's
of the p a r a m e t e r
mension
exists
does
conclusion
and it is valid
subset
there
since
X*
and
165
lies
in
H., l
nitely m a n y yield
if
M
hyperplanes
another Since
point
M
and
Vi
(2.8)
Vi, M
M ,
lie
in
because
Sp
(2.8)
holds
Indeed,
they
have
be a s s u m e d Now,
in t h i s
every
(2.9)
or that
are
linear
case,
there
is no s u c h
In c h a r a c t e r i s t i c
at
Let
be t h e c o n e
in
l-parameter
family
V
trivial
varieties X
(x,y)
and
are c u r v e s .
touching
V ,
if ~3
Then as
infinite.
So
M = Hi .
also
loci
number
but
have
XM
and
of points
if
X
Li = Hi
Vi, M
and
second
in c o m -
(= M)
the m
underlying
indeed,
transverse
it m a y
in
are r e f l e x i v e ;
irreducible
is p r i m e m
T
curve,
of r e f l e x i v e the theorem by
to
distinct
then
this
~ (V) % 0
Vi
if a n d o n l y
scheme
is so;
contact
M . hence,
contact.
at
= I ,
over
a second
(ii) t h e r e d u c e d
where
zm
will
1
are d i m e n s i o n a l l y
is t a n g e n t
touches
V
q ~ 2 ,
y = x mq+1 , T
would
(1.2).
underlying
Vi, M
line
of
and
and
tangent
H
M-contact
a finite
Qi ~ Pi
Vi, M
equation
by
obviously
the two
and
and
(Example. affine
X
XM
XM
that
~ 2
Hi
+ d i m ( V i , M) _< N-I .
where
is o f d e g r e e
infi-
> N-I .
only
by virtue
in
be
be
Hence
element
holds
would
then by symmetry
+ dim(V[)
that
(i) e q u a l i t y
Sp
Replacing
would
+ d i m ( V i) > N-I .
It is n o w e v i d e n t (Qi,Li)
then there
M .
a hyperplane,
d i m ( X M)
Therefore,
one
Thus
is f i n i t e .
(2.9)
XM
Sp .
holds. and
unequal,
containing
dim(X*)
In a n y e v e n t ,
if
of
were
are r e f l e x i v e ,
(2.8)
with
H
dim(X)
X
mon
H. 1
is t h e r e f o r e
(2.7)
If
and
surfaces implies (1.8)
q ,
in
that
indeed,
touches
let
X ~3
if
T
at
(zx,zy).
vary
in a n o n -
whose
there
and there
curve with
is s u c h t h a t
points;
also and
plane
is at
dual least
is, o b v i o u s l y ,
166
at least V
will
one have
The ond
X*
at least
general
contact
out
set
a general
m
of the
element
Let
S p+I
be
contains
t
in
T
case
in w h i c h
distinct
another
every
X
touching
copy
X
and
from
of
Vi
have
(Pi,Hi) Then
Vi .
a sec-
is c a r r i e d the
corre-
the p o i n t
((P1,HI),.-.,
where
Moreover,
contacts.)
(Qi,Li)
Vp+ I
(2.10)
line.
distinct
analysis
with
as follows.
sponding
meeting
represents
(Pp,Hp),
X .
(Qi,Li),
The p o i n t
t)
(2.10)
lies
in the
inter-
section
(2.11)
S p+I
where
J
is one
diagonally
on the
Of course, (Qi,Li)
H.
so,
J
if
a second = L. ,
1
i-th
Pi
and
already
A u t ( ~ N ) xp
then
been
acting
on
I x(p+1)
factor.
diagonal
In fact,
7 orbits.
(J x T)
of
(p+1)-th
= Qi ,
of the
has
7 orbits
is not the
are d i s t i n c t .
proper; not
of the
D
orbit,
because
Pi % Qi "
Indeed,
H i = L i"
Thus,
The dual
treated.
case,
Hence
the
(Pi,Hi) two
and
contacts
in p a r t i c u l a r ,
in w h i c h
4 orbits
Pi % Qi
remain
are J
is
but
to be con-
1
sidered. By the that
the
(Pr1-1CV1)D...D(prp-ICVp)
J xT
dimensions (2.11)
in s u b s e t s were
less
is empty.
ing n o n e m p t y ,
transversality,
it m a y
would
is not There
the o p e n are
D ( p r p + 1 - 1 C V p + I)
of c o m p l e m e n t a r y
than
dimensions.
complementary,
If t h e y w e r e
nite. In p a r t i c u l a r , J
of d i m e n s i o n a l
be a s s u m e d
sets
(2.12)
meet
theorem
more
than
be of d i m e n s i o n therefore,
by the
then
and
Indeed,
it c o u l d
complementary,
> I ,
and
so
F p+I
Sp
if the
be a s s u m e d then
that
(2.11),
would
less-than-complementary
be-
be i n f ~ case,
orbit.
2 similar
orbits
of c o d i m e n s i o n
I .
If
J
is the
one,
167
then
Pi £ Li
Qi £ Hi " sets
of
but
Say that
(2.12)
each
Qi ~ Hi ;
if
J
first.
is the
have
contained
in the c l o s u r e
reducible
components
and
such
that
is the
of
J xT .
and
X"
other,
By r e a s o n
an i r r e d u c i b l e
X'
every
J
such
X ,
that
((P,H) ,
(Q,L))
(CX')
Tp V i c L i . S i n c e the c o n t a c t at Pi i Hence Qi 6 H i , c o n t r a r y to h y p o t h e s i s .
Suppose,
nor
and
has
and
two
ir-
Qi 6 X"
X' c L i .
x (CX")
Similarly,
one,
Therefore,
6
P £ L.
is not
(2.10)
for e x a m p l e ,
satisfies
J
the two
containing
Pi 6 X'
but
point
(2.13)
Hi = Li .
Pi ~ Li
of d i m e n s i o n ,
component
Hence
then
the other,
finally,
of t h e s e
that
J
Consequently,
Tp X c L i . 1 is p r o p e r , t h e r e f o r e Thus,
after
all,
2 orbits.
is the o r b i t
of c o d i m e n s i o n
2;
in o t h e r
words,
(2.14) Then,
Pi 6 L i, Qi £ Hi by r e a s o n
first, the sure
has
an i r r e d u c i b l e
closure of
F p+I
of
J xT ,
J xT This
case
the
First, Pi 6 V'
roles
(2.16)
has
two
and
sets
and
such
Q E H .
V' c
of
(T
X
6
every
and
of
(2.12),
F p+I ,
meets
the
and of c o d i m e n s i o n it a p p e a r s
are
that,
in
cloI
in
if the The c a s e
interchanged
is similar.
V'
such
and
point
(CV') x (CV")
that
V"c
say the
and c o n t a i n e d
is not r e f l e x i v e .
(2.12)
It f o l l o w s
V") Qi
until
Hi % Li "
(2.10)
components
that
(Q,L))
sets
now
(2.10)
2, t h e n
irreducible
((P,H),
P 6 L
set,
n o w be a n a l y z e d
two
two
containing
other
from
Pi # Qi'
of the
containing
of the
Qi 6 V"
(2.15)
satisfies
and the
is d i f f e r e n t
Vi
and
will
one
component
in a s u b s e t
characteristic in w h i c h
of d i m e n s i o n ,
but
(Tp V') i
.
V"
that
168
Therefore,
V'
and
Secondly,
X
V" has
are e q u a l two
and
linear.
irreducible
components
X'
and
X"
such
that the points
((P,H),
(2.17)
satisfying in fact, most
I
P 6 L
this that
subset
is c o n t a i n e d
NOW,
the
Q 6 H
has
contains
Consider
(2.18)
and
6
form a subset
the point map,
of codimension
component
((Pi,Hi),
D
at m o s t
I ;
of codimension
at
(Qi,Li)).
f: D ~ X' x X" .
Its
fiber
over
(P,Q)
set
{ ((P,H), (Q,L)) 6 I X 2 i H D
in g e n e r a l ,
(CX') x (CX")
an i r r e d u c i b l e
natural
in t h e
(Q,L))
Q ~ TpX.
(TpX+Q),
In fact,
L D(TQX+P)}
X" ~ Tp X .
.
Indeed,
otherwise
contact
at
1
T
Qi
X c H.. 1
proper,
Li = Hi ,
P ~ TQX . I
in
However,
V" c H. l"
contrary
Consequently,
X' x X"
and
D
Hence,
since
to h y p o t h e s i s .
by r e a s o n is t h e
the
Similarly,
of d i m e n s i o n ,
full
inverse
of
is
in g e n e r a l ,
f(D)
image
Qi
is o f c o d i m e n s i o n f(D)
in
CX' x CX" .
Hence
(2.19)
Q 6 TpX
Moreover,
f(D)
and
covers
P 6 TQX
for a l l
Indeed,
X' .
(P,Q)
otherwise,
6 f(D) .
X" c T p . X .
Similarly,
1
f (D)
covers
X" .
The dimensions X'
is less.
Since
of X"
X'
and
N Tp X
X" has
are e q u a l . codimension
Indeed, I
in
say that X" ,
that
of
therefore
1
Tp X c X" .
So
Pi 6 X'
N X" .
However,
Pi
is a s i m p l e
Q
lie in
point.
1
The f(D).
line
Indeed,
is a c o n e a generator
(2.20)
C .
joining suppose The
through
P
and
the
contrary.
tangent simple
cannot
spaces points
TQC
=
Then the TQC
of
(Tp X)
are
C .
n
X
fiber
for all of
constant
Since
(TQX) ,
f(D) as
Q
(P,Q)
in
over
P. i
runs
along
169
the c o r r e s p o n d i n g
(2.21)
(V'NTpI. X)
In e i t h e r finitely and
case, many
rotate
in a pencil.
(V'NTQIX)
it follows
or
that
points
R
Now,
either
(V'DTp.X) ~ (V'D (TPiX+TQi X) ) " l V'
touches X
of
that
X
at every one of the
lie on the
line
joining
inP. 1
Qi"
dim(X')-1
,
let
meets
but C
E
point
C
and by
are distinct,
(2.19)
case,
Hefez-Kleiman general
hyperplane
V. 1
(2.22)
variety lence on
(2.23)
to e s t a b l i s h
For V
i = 0,...,N-I
is reflexive,
of I ,
~N, by the
[CV]
X
the class
1
[CV]
is equal Y'
and
then 2 .
Y" lines;
contacts. (see
so is a Hence,
if the
is not reflexive.
there
in the
this n u m b e r
A
G
if
is not
G = E N TpX
It is not hard to prove
of points
let
line
two d i s t i n c t
in c h a r a c t e r i s t i c
following
P
set of tangent
remains
set
is given by e v a l u a t i n g
the
the
Thus,
line makes
from 2 , then
Since
over
Obviously,
they have the same
except
Y" = E N Y" . Let
f(D)
Hence
TQY" .
is equal
SPrI*[CVI]...Pr2*[CVp]
it suffices
Lemma.
to
and
of
P .
the p r o o f of the theorem,
multiplicity,
of the
from
if a v a r i e t y
is d i f f e r e n t
C
TpX .
are not reflexive.
to show that the n u m b e r
with natural
Y' = E n X'
in
it is equal
section,
space of c o d i m e n s i o n
fiber
then every t a n g e n t
they
To complete
modules
The
distinct
[1983])that,
characteristic
task,
Y' .
then by s y m m e t r y
if they coincide, In either
of
Q
linear
the curves
is a h y p e r s u r f a c e
in a point
TpY'
be a g e n e r a l
and c o n s i d e r
be a general
a cone,
to
=
simple
Finally,
P
TQX
Sp
only one m a j o r of
(2.5),
the p r o d u c t
counted
of the
to
[F p] ,
lemma.
be an i-plane. is given,
Then,
modulo
formula
= r0(v) [CA 0] + . . . + r N _ I (V) [CAn_ I] .
for any sub-
rational
equiva-
170
Proof. ri(V)
The and
formula the
following
(2.24)
form the
wi
a basis
follows
classes
[CA i]
f o r m the
is a p r o j e c t i v e - s p a c e more,
(b)
says,
2 facts:
in o t h e r
1
(2.25)
results
In any given
l-parameter
variety
V
r(v) ;
and
Theorem.
(a,ii)
furnished
of the by
N
this
standard
Fix a c o m p l e t e
the e l e m e n t s
I ;
Now,
(b)
for
(a) h o l d s bundle
i = 0,...,N-I because
~N x~N
I/~ N
.
Further-
(the Kronecker function) .
13
from
Views
(1.5),
(1.6)
and
(1.7).
and A p p r o a c h e s .
the
number
position fact
in
Conversely,
in w k i c h
of v a r i e t i e s ~N
is the
case
p = I
and
touching
the
of P a r t s
is a g e n e r a l
linear
a
is f u r n i s h -
n o w be p r o v e d ,
families,
B
X
is f i n i t e
as w i l l
l-parameter
flag,
on
trivial
= 6..
family,
ed by the m o d u l e
of the
that
directly
in g e n e r a l
i = 0,...,N-1
basis.
of the
3
3. O t h e r
for
(1.3)
Pr2,ci(0(I ))N-l-i
dual
words,
r (A)
f r o m the d e f i n i t i o n
of N - c y c l e s
subbundle
(2.25)
Finally,
(a)
= Prl,c1(0(1))i
for the
elements
immediately
(a,i) r(V)
is
pencils.
j-plane,
3 (3.1)
For
~ = B_I c B 0 c . . . c B N = ~ N
i = 0,.--,N-I
rametrized
by
consider
T = ~I
,
of all
(3.2) The
of the
(3.3)
linear
pencil,
(N-1-i)-planes
the X
natural
family
pa-
such that
B N _ i _ 2 c X c BN_ i .
number
Pr1:
the
.
of
X
projection
I xT ~
I
touching formula
obviously
a general applied
carries
to F
I
j-plane (2.22); onto
A. 3
is
indeed,
6.., z3 the
the v a r i e t y
W i = {(P,H) 6 IIP 6 B N _ i _ 2, H = B N _ i} ,
by v i r t u e
projection
171
and the class [CAi].
[W i]
obviously
Since the number
r (V),
it is just Segre
([1912]
the ranks;
namely,
planes
that
X
§37, p.924) r0(v)
a general
of a general
intuitively
obvious
Schubert
presented
of
because
and d e v e l o p e d
in two r e s e a r c h
may be expressed
pressed
series
as a sum of two others
of e n u m e r a t i o n
to
tive
of i m p o s i n g
condition
(that is,
From Schubert's as follows:
point
the c o n d i t i o n
mily to t o u c h
to t o u c h one of
r. (V) l
may be c a l c u l a t e d linear p e n c i l
in its proof
one of
r0(v)
V
points,
and of
r
1
the
V . This
as a proof
(1876,7)
by
of problems;
when
In he
and then made
Schubert's
it is that
is
formally.
idea of dual bases;
linear
(V)
point
a given
combination
it is e q u i v a l e n t
of
enumerative of s t a n d a r d
([1879]
p.282),
by i m p o s i n g
a condition
it
the
may be ex-
for the p u r p o s e s
the same numbers
as)
the a l t e r n a -
one or the other. of view,
r0(v)
the e n u m e r a t i v e
V
points,
touch
V .
varying
r1(v) of
results
etc.;
look
in a l-parameter
lines,
etc.;
(N-1-i)-planes the Lemma
on an element
to the c o n d i t i o n
lines,
above
to the a l t e r n a t i v e
Similarly,
the c o n d i t i o n
is e q u i v a l e n t r1(v)
X
is e q u i v a l e n t
as the number
look as follows:
long to a v a r i e t y
proved
of
of hyper-
the e q u i v a l e n c e
may be c a l c u l a t e d
on a v a r i e t y
that
section
"characteristics"
it yields
a given v a r i e t y
dition
general
he t e r m e d
coefficients
in a dual
V ,
loosely
Of course,
as a formal
which
and that the c o m b i n i n g given c o n d i t i o n
linear
articles
In essence,
down-to-earth.
conditions,
description
it is based on the use of dual bases.
view
or basic
and t o u c h
and is e a s i l y
[1879].
condition
dual to
by e v a l u a t i n g
V - is the number
one are equivalent;
the grand theme of his book is more
(2.24),
interesting
lemma may be d e s c r i b e d
introduced
it first
of
is f u r n i s h e d
another
(N-2)-plane
geometrically
The p r o o f of the
fact,
V
i-codimensional
and the p r e c e d i n g
calculus,
gave
- the class
description
Schubert
touching
wi
r. (V). 1
contain
is the class
of
is the e l e m e n t
facon-
furthermore, X
in a and the bases
(P,H)
to be-
on it to belong
and the c o n d i t i o n s
on
to
(P,H)
172
to belong to an i-plane for that
P
lies in a
Schubert
(N-i-2)-plane while
([1879] pp.50-1,
number of v a r i e t i e s ty
V
in
i = 0 ,...,N-I
~2
X
~3,
I ,
contains an
(N-i)-plane.
derives the e x p r e s s i o n
for the
family t o u c h i n g a given varie-
via a v e r s i o n of the Lemma,
from the canon-
e s s e n t i a l l y as was done in Section 2. However,
he obtains the bases differently. but d i s t i n c t ways,
H
in a l-parameter
and in
ical dual bases on
289-95)
are dual to the conditions
In fact, he p r o c e e d s
in two similar
starting in both cases from the c o r r e s p o n d e n c e prin-
ciple.
One way the d e t e r m i n a t i o n of a K~nneth d e c o m p o s i t i o n of the rel-
ative
d i a g o n a l of
d i a g o n a l of this work).
I.
I / ~ N , and the other
(See Grayson
[1979]
involves that of the absolute
for a lovely up-to-date v e r s i o n of
Schubert's m e t h o d was years ahead of its time and is still
rather interesting. Schubert
[1879] goes on, at least in the specific cases he con-
siders, to give the e x p r e s s i o n for the number of family t o u c h i n g given proceeds
X
in a p - p a r a m e t e r
V I , . . . , V p . He does not fuss, but implicitly he
in a somewhat d i f f e r e n t way from that in Section 2. Essentially,
he forms for each
(3.4)
V
the d i v i s o r i a l cycle on
ZV = Pr2,((Pr1*[CV])
T ,
[FI]) ;
its u n d e r l y i n g set is the closure of the set of all points of r e p r e s e n t an [CV]
X
touching
V .
By additivity,
yields a similar formula for the class
the number of
X
touching
(3.5)
V1,...,V p
the formula [ZV].
Now,
(2.23)
is equal to
follows immediately.
must be locally principal,
the cycles
[ZV i]
the cycles
must intersect prop-
erly , and the i n t e r s e c t i o n m u l t i p l i c i t i e s must be investigated. T
for
in principle,
For the p r e c e d i n g argument to be rigorous and complete,
if
that
IT[ZVI]...[ZVp]
The desired e x p r e s s i o n
[ZAj]
TO
is r e p l a c e d by a smooth, b i r a t i o n a l l y
e q u i v a l e n t variety,
Now, then
173
any d i v i s o r i a l that there
cycle on
T
is an induced
will
action of
only finitely many orbits these Then, bit,
given their
several
intersection
by orbit, Schubert
(1874-9),
al c o n d i t i o n s Earlier,
to prove
introduced
p
p = 1 ,
and c o n s i d e r
p = I ,
equal
(3.6) Ii fix
The number
Ii ,
no or-
multiplici-
transversality
(1873),
results,
developed
method,
the power-
imposition
that d i s t r i b u t e s in w h i c h
of sever-
over
sums.
the several
to use this m e t h o d
in a p - p a r a m e t e r in general
family
position
r(V0)...r(Vp), subfamily
X
is fur-
assume
of those
X
touching
touching
V1
a general
(p-1)-parameter
of these
X
the case
touching is, by the
touching
The d e s i r e d the
of a p p e a r a n c e
i-plane
subfamily
by induction,
one must define
the m u l t i p l i c i t i e s
of those
V2,...,V p furnished result
of the
X
X
is, on the by the p r o d -
follows
subfamilies
Li .
imme-
with enough can be con-
and compared.
De J o n q u i ~ r e s curves but
the
r(V2)-..r(Vp).
To be rigorous,
so that
trolled
V ,
each contains
intersection
in the s u b f a m i l y
and on the other,
of the modules
diately.
plane
cases.
to
and c o n s i d e r
Li .
one hand,
care
l-parameter
is the number of these i ,
touching
uct
X
VI,...,V p
X
it has
important
For example,
of v a r i e t i e s
of
suppose
r0(Vl)l 0 + o.. + r n _ 1 ( V 1 ) I N _ I ,
where Now,
by a p r o d u c t
successively.
the
and that
simultaneous
of their m o d u l e s
V 2 , . . . , V p . The n u m b e r case
by H a l p h e n
a cumbersome
given v a r i e t i e s
by the p r o d u c t
and the
namely,the
used
that the n u m b e r
touching
such that
Moreover,
of the subvarieties.
is r e p r e s e n t a b l e
are
T
T
in some
the general
inspired
(3.5);
he and others
conditions
nished
in
on
obtain
will be p r o p e r
to the traces
ful idea e m b o d i e d
of
by a p p l y i n g
principal.
A u t ( ~ N)
conditions
subvarieties
ties m a y be q u a l i f i e d orbit
be locally
X
(1861)
gave
a simple
in a l-parameter
it turned out to be v a l i d
family
expression touching
in a r e s t r i c t e d
for the n u m b e r a given plane number
of
curve
of cases only.
174
Cremona
([1862]
expression
greater;
of a general
mona and Chasles Zeuthen alized
111 bis a.,
by i n v o l v i n g
did s o m e t h i n g part
Nr.
(1871)
lowing
Thm.
family;
dence p r i n c i p l e
19th c e n t u r y
"principle
It was k n o w n
by Schubert,
1874)
by Schubert,
1876).
each Now,
Vi
between Hence,
CX
and
provided
degenerates matter),
CV, 1
appropriate
when
V. 1
does
abstract
equivalence,
applied
One c o m m o n l y
fact, to
homolography a fixed point
is the P ,
of the gener(1866) Coolidge
proof,
fol-
parameter
the c o r r e s p o n -
on w h i c h
there
of c o n t i n u i t y
of special
attention
V. 1
V. 1
remains
(so
(so named of
the same w h e n
of d e g e n e r a t i o n . to an i n t e r s e c t i o n
induces
a motion
is paid to the way
of
CV. 1
in w h i c h
CV. 1
be said b e l o w about
of the p r i n c i p l e
that a l g e b r a i c
were
that the number
corresponds
of
major
(so named
of number
for example,
(and more will consequence
position
of c o n s e r v a t i o n
VI,...,V p
is another
many e n u m e r a t i o n s
even to the point
and a
equivalence
this
is a c o n s e q u e n c e implies
numerical
(2.22).
used m o t i o n
of d e g e n e r a t i n g
Cre-
by Cayley
reuses
as
principle.
the case of a several
principle,
says,
and a m o t i o n
then the present
of the more
family
continuously, X
De Jonqui~res,
Zeuthen's
as the p r i n c i p l e
touching
an
of m o d u l e
step.
The p r i n c i p l e
between
presents
the p r i n c i p l e
family
independently?)
independently.
but he u n n e c e s s a r i l y
and the p r i n c i p l e
is varied
a contact
and himself
he treats
successively
in a p - p a r a m e t e r
notion
w h i c h was d i s c o v e r e d
of geometry",
1822),
(1864,
He used his own v e r s i o n
16, p.440)
induction
de Jonqui~res'
on the c o r r e s p o n d e n c e
(1871)
Then
Chasles
the a b s t r a c t
to the c o r r e s p o n d e n c e
named by Poincelet,
X
Thm.
as above,
at the
In a d d i t i o n
based.
by Brill
Lehrbuch.
he argues
proof.
principle,
refined
of enumeration.
their p r o o f s
14, p.436;
Zeuthen's
r(V).
theory
another
and w h i c h was p r o v e d
I, p.170)
he i n t r o d u c e d
based
correspondence
([1959]
the m o d u l e
self-dual
gave
Thm.
has been the
automorphisms
family whose
of
~N,
of d e g e n e r a t i n g axes
flow p r o v i d e d
are a fixed
by a c e r t a i n
the homolography.
homologies,
whose
complementary
The
centers
hyperplane
are H
175
and whose
cross
homologies [1879]
degenerate
(Ex.
principle number
curves
and obtain
classes
way;
of
level, not
this
center
P
form
for an a p p r o p r i a t e
CW
I , whose
a hypersurface,
then
proceed
and t a k e
ography
within
that on
I
bination
of the
the
The combining zation
conservation There Schubert a formula after
[CV]
cycles
is a t h i r d
[1903] of
special
Porteous
treatment
The third is i r r e d u c i b l e ,
proof
this
(2.25).
proof
of t h e
be
first
one
generate
[1905], Today,
general
as f o l l o w s .
say of dimension
~N
in an the
H .
itself. in
leads
H
is a j - p l a n e ,
was
[1974]
d . Next,
is
a homol-
conclusion
for
com-
j=0,...,N-I.
characteri-
on a formula known
Porteous's
of
of
of as
formula,
for an a l g e b r o - g e o m e t r i c
of wide
First,
V
the p r i n c i p l e
originally
it is c a l l e d
If
with
of the
It is b a s e d
which
of the
to a linear
section,
a
At a n y rate,
to the
by virtue
not
into
are e a c h
axis
([1974]
does
degenerates
equivalent
of t h i s
formula
of
contained
r. (V) 3
lemma.
V
the
H
Aj
(See K e m p f - L a k s o v
runs
is p r e s e n t e d
of M a c P h e r s o n
CV
of
procedure
where
and
o f an e v e n m o r e
this
and an i n d e p e n d e n t
components
W
will
beginning
position.
[1971].
then
is r a t i o n a l l y
and Giambelli
proof
[CAj]
construction
properly
are the
at t h e
of n u m b e r
W
[CAj] ,
coefficients
plane
to f o l l o w
f r o m the
its c h a r m
irreducible
W
cycle
the
for the
a given
0 ,
the
a n d the
expression
touching
if a s u b v a r i e t y
one of the
Repeating
of t h e m g i v e n
that
subvariety
H .
Schubert
a homolography the
the
content.
reduced
care of the
H .
differs
of a h o m o l o g r a p h y ,
of
to
Lemma;
However,
0 ,
subscheme
P
family
fact
the graph
in c h a r a c t e r i s t i c
the
proof
I .
using
r ~ 0 ,
to e s t a b l i s h
of the
use the
on
as
in c h a r a c t e r i s t i c
proof
in its g e o m e t r i c
It c a n be p r o v e d
contain
at l e a s t
r ;
from
uses
in a l - p a r a m e t e r
it d o e s
lie
that,
of number
(N-1)-cycles
importance
the p r o j e c t i o n
a second
On a technical
number
for e x a m p l e ,
of c o n s e r v a t i o n
important
a variable
into
It is p o s s i b l e ,
approach
§5)
are
4, p p . 1 3 - 1 4 ) ,
of plane
curve.
next.
ratios
applicability.)
it m a y consider
be a s s u m e d
that
V
the Nash modifica-
176
tion,
f: V'
f(P)
~ V .
is simple,
of the
graph
its t a n g e n t locally
A point
then
T
free
of
is
of the G a u s s d-plane.
P
On
V'
represents
Tf(p)V ;
map, V'
which
there
indeed,
carries
are two
0 ~ K ~ 0V ,N+I ~
(3.8)
0 ~ A ~ B ~ f*0v(1)
~(B)
inclusions by
V'
T-;
is just
a simple
important
point exact
if
the
closure
of
V
to
sequences
of
sheaves,
(3.7)
In fact,
a d-plane
is the
T c ~N
total
are
space
specified
B
~ 0 ,
~
0
of the by
family
(3.7),
of d - p l a n e s
and the
T ; the
inclusions
P £ T
(P,H)
that
(3.8). Let
f(P)
I'
£ H .
It is e a s y
denote
Consider to
the its
u:
and w h e r e from
P2
Pl S
subvariety
see t h a t
(3.9) and
to
Z
P2
P2
Pl B
is the
S ~ Pl are the
lands
(3.10)
[Z]
by P o r t e o u s ' s
subvariety
formula.
in
of
V' x ~ N
Z
of t h o s e
locus
A
by
(3.9)
such
such
that
TcH
.
of a m a p
S = 0~N(-I)
projections; Pl A
(P,H)
zeros
where
indeed,
because
= cd(P 1 A - P2
So,
of
of
the
P 6 H .
natural
map
Hence
S)
there
is a s h e a f
C
on
I
such
that
(3.11) where ally
[Z] p:
I' ~ I
onto
CV .
is the Hence,
= cd(P I B-p
C)
natural
Obviously,
expanding
map.
(3.11)
and
using
p
carries
Z
the p r o j e c t i o n
biration formu-
la y i e l d s
(3.12) where
[CV] s i ( C v)
If
V
is the
= ~i P * P 1 * C d - i (B) i-th
is a d - p l a n e
Segre
class,
A d , then
B
" si(cV) or
inverse
is t r i v i a l
Chern and
class.
(3.12)
yields
177
(3.13)
[CAd ] = Prl
ci (01pN(1))N-d
. Sd(CV ) , *
where
Pr1:
I ~ ]pN
lish the lemma,
is the p r o j e c t i o n .
it r e m a i n s
(3.14)
f,cd_i(B)
Now,
theorem
by B e z o u t ' s
Since
w
P*Pl
= Prl
f* '
to e s t a b -
to p r o v e
= ri(V) c I ( % N ( I ) ) N-I .
and the p r o j e c t i o n
formula,
(3.14)
is e q u i v a l e n t
to
(3.15)
ri(V)
H o w e v e r,
it is e a s y to see t h a t
(3.16)
Cd_i(B)
Since tion
= ~ C d _ i ( B ) f * c I ( 0 V ( 1 ) ) i [ v '] .
Z ~ CV (1.3)
= Sd_i(KV)
is b i r a t i o n a l ,
of
r. (V) 1
Z = ~ ( K v) . Hence,
therefore
of
d
V.
is the m i n i m u m Let
n-plane
in
In ~N
the set of p a i r s
denote and
P 6 L . Let
(P,L)
such that
and p u s h i n g
the r e s u l t
follows
as follows. of the
from the d e f i n i -
Fix
n ~ N-1-d ,
irreducible
(P,L)
denote
P
(2.23)
% )N-1-i i ( N (I)
of p a i r s
C V n
(3.7)
formula.
of the d i m e n s i o n s
L + T p V % ~ N . The n p u l l i n g (P,L,H)
easily
the v a r i e t y
*c
(3.15)
and the p r o j e c t i o n
The lemma m a y be g e n e r a l i z e d where
= p l , P2
by
where
the c l o s u r e
L in
is a s i m p l e p o i n t of
up to the
d o w n to
I
n
components is an I
of
n
V
and
(flag)
manifold
of t r i p l e s
yields
the a n a l o g o u s
formu-
la,
(3.17)
[CnV]
because
the i n v e r s e
while
if
inverse
W
[CnV]
i m a g e of
is a s u b v a r i e t y
i m a g e of
Formula
= rN_1_n(V) [CnAn_1_n]+...+
CW
(3.17)
is a s o m e w h a t
drops
CV of
projects ~N
dimension
under
because,
special
on
[CnAn_ I] do not g e n e r a t e
the c l a s s e s
birationally
of d i m e n s i o n
is r e m a r k a b l e class
rN_I(V) [CnAn_ I] ,
In .
onto
< N-1-n ,
the p r o j e c t i o n for one thing, Indeed,
of c o d i m e n s i o n
to
CnV , t h e n the I
n
it says that
[CnAn_1_n],-.., n+1
for
n < N-I .
178
However,
they
are m e m b e r s
for all
the
classes.
a basis
was
found
20),
and more
group but
and
[1974]
([1958]
§5),
obtained
cases,
almost
all
work
or
next
The other
of t h e
proof
of
I
by Ehresmann
G
(1953,
to M a r t i n e l l i
and
all
Using
directly.
n = I
and
(a,ii) X
or
case;
([1934]
(1942),
this
algebraic
on
Zobel
in t h i s w a y
Theorem
is a c u r v e
(More w i l l
§§19,
basis
d = N-I ;
of t h e
such
unpublished
In fact,
(ii)
VI
is a h y p e r s u r f a c e .
basis
is a r e d u c t i v e
by Chevalley
(3.17)
(a,i)
is a s p e c i a l
n
Cor. (a), p . 6 9 ) .
(i)
almost
on
self-dual
corre-
for and
be s a i d
p = I (ii)
about
Zobel's
section.)
two proofs
of t h e
(In t h e
third
Grassmannian
(3.15),
following
Parts
VI
mutandis.
sheaf
cases,
natural
where
subgroup,
the basis
(i) e i t h e r
X
in t h e
mutatis
in t w o
he proved
in t w o
G/P
h a r d to e s t a b l i s h
(3.17)
spondingly,
basis
Prop.1(a),
attributing
lovely
flag manifold
on any
is a p a r a b o l i c
it is not
this
on any partial
see D e m a z u r e
In ,
In fact,
generally,
P
of a rather
etc.)
even more
lemma
also yield
proof,
take
of n-planes,
In fact,
general
these
setting,
S
directly,
to be t h e
although,
two proofs where
(3.17)
universal
of course, work
not
as w e l l
the proof
using
sub-
in t h e
in t h e
dual
bases
fails. In t h e
general
irreducible
variety
the
Grassmannian
setting, Y
~N
is r e p l a c e d
of dimension
bundle
G n ( ~ $) ,
N.
Correspondingly,
which
where
P 6 Y
and
L
is an n - d i m e n s i o n a l
space
TpY ,
and
S
denotes
I
n
is t h e
ducible CnV P
denote
Id .
point
of
but
map, The
which
S(I)
in
In
V
and
c a n be o b t a i n e d carries
sequences
V
vector
earlier
of
where
and
Y
# tpV .
the
closure
point
(3.8)
P
of
denotes
the pairs
if
Y = ~N For
(P,L)
are r e p l a s e d
let
such that
modification
of the graph V
• then
an irre-
d > N-1-n ,
The Nash
(P,L)
of the tangent
S .
set o f p a i r s
L + tpV
a simple
subbundle;
smooth,
In
subspace
is t h e
of the
by taking
(3.7)
parametrizes
its u n i v e r s a l
subvariety
the closure
is a s i m p l e
"Gauss"
as b e f o r e
d-dimensional
f: V' ~ V
of
same
b y an a r b i t r a r y
to the
of the
pair
by a simple
(P,~V) sur-
179
*
jection, on
V'
1
f ~y ~ ~ , induced
A = ~(I).
where
by the
The c l a s s e s
of t h e
the Mather-Chern
lemma
position.
Finally,
consequence trivial;
generalize
of
statements If
then,
Part
and the
Proposition.
least
CnW
is n o t
W
then
[CnW]
n
through
runs
,
V .
The
second
and third
proofs
(a),
(b) o f t h e
following
pro-
(c)
is a s i m p l e
= [V] ; t h e of
second
mi
to
V0
whose
described
in a f l a t zero,
C V n
irreducible subvariety
scheme
a set o f
of dimension
through
is a c l a s s
and of
formal
assertion
is
(3.7).
above,
V0 ,
family
the
of
following
W
[CnW]
subschemes
degenerates components
W
of
then
W
of
a basis
subvarieties
correspondingly
Moreover,
is c o n t a i n e d
i
on
for t h e
Y
whose for
classes
of dimension
as a l i n e a r is an i - p l a n e is
ri(V)
.
i
on
combination for
Y .
if
W
in t h e
classes
form a
i = N-1-n,...,d of c o d i m e n s i o n
, n+1
v)
In p a r t i c u l a r ,
of the
i = N-1-n,.--,d
[CnW]. ,
Y ,
are e a c h o f t h e
V0 .
L~r~-1-nS0(SV) +'" • + m d S d + 1 _ N + n ( S
y = ~N
then
y = ]pN
form
c a n be e x p r e s s e d
cient
setting
d
V,
of Part
md(V)
if
of rank
V0 .
(3.19)
where
,
of the
classes
of the
n
general
Id ;
sheaf
= m N _ 1 _ n ( V ) s 0 (S v) + . . . + md(V) S d + l _ N + n ( S v) •
runs
for t h e
I
I
of
[CnV]
basis
on
of
locus
As
Parts
is a r e s t a t e m e n t
for an a p p r o p r i a t e
(b)
(c)
of
assertion
in c h a r a c t e r i s t i c
a component
singular
and yield
is d e g e n e r a t e d
to a s u b s c h e m e form
classes
(b), b e c a u s e
In t h e
on
= f,cd_i(~)
first
third
i
on
free
hold.
V
at
the
locally
quotient
of dimension
mi(V)
are called
is t h e
universal
(3.18)
(a)
~
then
[CnV]
Moreover,
if
the coeffi-
180
4. O p e n p r o b l e m s .
There finement sents
are
of t h e
condition
in S e c t i o n
in w h i c h
possibility
case,
first they
such
contacts
more
precise
a bitangency case.
space
a hyperplane,
but
not
property: any rate
there
sists
there
In t h e points over,
and the second
of contact, since
both
are d i s t i n c t , the common the
tangent was
made
see t h a t , proper make
above
a second
components, and,
case,
does
other
that
a linear contact,
spaces
linear
components and
since
the dimension
of t h e
a variety
proper
the
two
of t h e o t h e r
not proved
points
lie e n t i r e l y
are p r o p e r
at t h e p o i n t s
with
not
both
of course,
and t h e
dimension
but
simple
two two
is
components
of c o n t a c t in S e c t i o n that space
enjoys
one of the
equal
but
at
of these
subset
the
that
two con-
the tangent line
joining
in t h e v a r i e t y . component contain the
contains one each.
hyperplanes linear
is
> 2
Conversely,
the
above
point
More-
of contact plus
it f o l l o w s
and that
their
(This a s s e r t i o n
2.)
of c o m p l e m e n t a r y
both
component
Similarly,
are d i s t i n c t .
at a d i s t i n c t
second
following
such that
of t h e
> N .
is t h e
is a l i n e a r
the
contain
narrow
possibility.
case,
possibly
closed
Vi .
analysis
in t h e
that
has
family
are d i s t i n c t .
in t h e p r o d u c t
locally
pre-
is o n e
this
second
variety
the
and there
are e q u a l ;
a component
re-
varieties
there
of contact
in t h e
which
Indeed,
is t o r e s o l v e
other
> 2 ,
are d i s t i n c t ,
contacts
whenever
contact
and the
of d i s t i n c t
line
dimension
spaces
has
irreducible
it f o l l o w s
common
V. l
is a 1 - c o d i m e n s i o n a l
at t h e p o i n t s
the points,
or
given
occur,
of contact
a direct
in t h e g i v e n
Namely,
does
Furthermore,
same dimension
o f all t h e p a i r s
spaces
that
are t w o
of the
components,
X
X
necessary.
the points
hyperplanes
varieties
not
mean
(c,iv),
one of the
The problem
the
are d i s t i n c t .
to be no
actually
case,
would Part
information.
of b i t a n g e n c y ,
two bitangent
concerns
with
but
cases
too
solution
for t h e r e
is c o n v e n i e n t
of a second
In b o t h In t h e
proper
2 yields
whose
One p r o b l e m
condition
two distinct
Now,the
problems
Theorem.
a sufficient
making
case
several
it is e a s y t o
property,
makes
dimension, and with
a
it m u s t
a distinct
181
hyperplane variety
of contact.
that enjoys
The problem,
the p r o p e r t y
then,
is to find an e x a m p l e
or to p r o v e
of a
that no such v a r i e t y
can
exist. Another asserts
problem
only that
already.
Katz
is to s t r e n g t h e n
each contact
[1973]
proved
family of h y p e r s u r f a c e s general
position
quadratic provided
singularity that
and either Ferrarese teristic
(i)
those V
= I
the c h a r a c t e r i s t i c
is
0 ,
(c,i)
that
in
touching
V
X nV
proved
is known
in a l-parameter
(and no other or
dim(V)
that,
if
which
V
in
with a nondegenerate
is r e f l e x i v e
is not 2 or
(pvt. com.)
~N ,
more
a given v a r i e t y
of contact
and
of the Theorem,
In some cases,
in a v a r i e t y
at the point
deg(X)
and Hefez
is proper.
essentially
X ,
intersect
Part
singularity)
(ii)
deg(X)
is even. N = 2
> 2
Furthermore,
and the charac-
then the
intersection
multiplicities
satisfy
i(P;X.V)
= i((P,TpV) ;CV.Pr1,[F1])- I ,
the re-
lation.
(4.1)
whether
or not
V
is in general
position.
The p r o b l e m
is to g e n e r a l i z e
these results. There
are several
T h e o r e m have away
directions
been g e n e r a l i z e d ,
from the Theorem,
issues
will
literature
cite more than These w o r k There
and their
between
is extensive,
references
has been work
in an e s s e n t i a l l y
the curves
(1,0)-th and
of several
result in two
(0,1)-th
several
way,
the
for surfaces l-parameter
characteristic
in
families numbers
recent
to
works.
information. be-
§14)
proved,
for plane
curves,
the points of plane
10,1 ~
and
However,
of contacts
([1879]
result
~3:
directions
for more
of e n u m e r a t i o n
following
step
be i m p r a c t i c a l
should be c o n s u l t e d
Schubert
one
indicated.
and some of the most
families.
cases of the
issues, These
and it w o u l d
in the theory
rigorous
a similar
are
special
and some open p r o b l e m s
some of the e a r l i e s t
tween the v a r i e t i e s
alongside
and there
various
on w h i c h w o r k has been done.
now be discussed,
the re l e v a n t
in w h i c h
and
of contact
curves 11,1 ~
with trace
182
a curve
of degree
(Ist class)
(4.2)
r I = 1016
and their
tangents
envelop
(4.3)
a curve
+ 1016 ,
of class
r 0 = I01 { + 1116
This w o r k
for c u r v e s
Pieri
(see
found
the
varying
number
families
Zobel
contact
spaces
did
It w o u l d
varieties three
like
or m o r e
swer the
"the p r o b l e m given
of
in i n t e r s e c t i o n a separate paper
was
cations, matter
have
systems
paper."
was
given
contacts,
number
not even
to
numbers he
at
P
[1958] in
~N
of parameters.
found
two varieties
contacts
with §6)
the
have
number
a k-dimen-
and their
had
to use
tangent
raises
be m o r e
(an o b i t u a r y
died young,
Zobel
He o b s e r v e d
available
at t h e
but not the multiplicities expected
values
of
that
time
I of
could
he w r o t e
here with
an-
in a
varieties
an important
and
of
this:
of varieties
including (Fn(6))
product
one
appropriate
notice,
[1964]).
significance.
studied
on
to t h e c a s e
products,
numbers
dimensions
Zobel
a fibered
the passage
kind
of the
an i n t e r s e c t i o n
in m i n d w h e n
for t h e
it w i l l
to t h e p r o o f
involve
fibered
of t h e
suitable
by Scott
similar
facilitate
Unfortunately,
the
Zobel
any dimension
general;
smooth
however,
expressions
and transversality
total
are
themselves
([1958]
which
not published
related
more
have proofs
Perhaps,
of
theory
of e n u m e r a t i v e
position of the
finding
of
work,
by
k-plane.
if o n l y
Zobel
system which
in t w o o r m o r e
if t h e y
proofs
families.
question
P
to h y p e r s u r f a c e s
In r e l a t e d
varieties
a little
seem better,
FP ,
121).
By d e f i n i t i o n ,
results
The original
generalized
of appropriately
contacts.
a common
was
Fn.
(0th rank)
+ 111~
between
something
The preceding
Ik .
[1915]
at a p o i n t
contain
Theorem.
surfaces
of contacts
of k - d i m e n s i o n a l s~nal
and
Zeuthen-Pieri
in t w o
In fact,
or
+ 1116
question
to d i s c u s s apparently
in that
a list of publi-
did broach
the
the
of general
yields
theory the
of a p p e a r a n c e
I in c h a r a c t e r i s t i c
finiteness of t h e 0 . Of
183
course,
currently
an open p r o b l e m There tiple
available
has been
§6)
employed points, curves
a lot of w o r k contacts.
said this:
osculating
system...;
these
of
to curves
given
s y s t e m and tangent
results
Now,
this
where
a curve
second
family;
(0,1)-th degrees
there
in one
by the
Zeuthen
found
(1879)
and surfaces
use of similar curve
in
proach
~N
this
his e n u m e r a t i v e
(1880)
and if
and o t h e r s
note
the
if
denote
of the
the conics
denote
formula
is
same type
for curves
on p.278
of number on the
cut or t o u c h
points.) It is an open p r o b l e m
the
of the
of c o n s e r v a t i o n
that
in a
(1,0)-th and
the classes
also the report
it
of points
a curve
kl,k ~
then the
and the p r i n c i p l e
p.274;
after
formulas.
osculates
families,
tangents,
interesting
denote
spaces
than a curve,
for the number
family
k0,k 6
given
of a point on
19th c e n t u r y
~i,~
of the
formula
other
to
a given
to put this
ap-
basis. recovered
theory
(4.4)
of triangles.
Roberts-Speiser
rigorously.
and
to e n u m e r a t e
at several
Schubert
theory
numbers
[1915]
on a rigorous
nouncement,
l-parameter
at the same
by c o n s i d e r i n g
more
Zobel
can also be
of another
two n e i g h b o u r h o o d s
formula
~0,16
are,
a variety
become
of mul-
or o s c u l a t i n g
to v a r i e t i e s
with
will
inflectional
methods
systems,
is an e x p l i c i t
via h o m o l o g r a p h y
(see Z e u t h e n - P i e r i
first
It is
that theory,
which
some
loci of cusps,
loci e n v e l o p e d
spaces
and g e n e r a l i z e
if
characteristic of the
the
general
namely,
contact
approach
has been used to recover In the plane,
with
may then be g e n e r a l i z e d
by c o n t a i n i n g
the variety."
[linear]
of e n u m e r a t i o n
to those of §§2-4
of two given
w h i c h have more than o r d i n a r y for instance
In c o n n e c t i o n similar
1's and more.
all this work.
in the t h e o r y
"Arguments
to find the n u m b e r
of one
y i e l d the
to clean up and g e n e r a l i z e
and h i g h e r - o r d e r
([1958]
t h e o r y will
However,
in p a r t i c u l a r Roberts
[1980])are
as an a p p l i c a t i o n
and Speiser
currently
it is an open p r o b l e m
working
(see the out
of
an-
Schubert's
to g e n e r a l i z e
the
184
t h e o r y to cover c o a l e s c e n c e De J o n q u i ~ r e s
(1866)
among more t h a n three points.
found a formula,
w h i c h has b e c o m e
the n u m b e r of p l a n e c u r v e s of g i v e n d e g r e e that m a k e c o n t a c t s of g i v e n o r d e r s w i t h a g i v e n curve appropriate
n u m b e r of points.
by V a i n s e n c h e r some r e l a t e d Vainsenoher
the v a r i a b l e
repeated m-fold
and the
Issues of this
formula may
is e n t i t l e d ,
calculus",
c o n t a c t s w i t h e a c h of the
lose its e n u m e r a t i v e
"Rigorous
and H i l b e r t wrote,
about H i l b e r t ' s
formula,
f o u n d a t i o n of S c h u b e r t ' s
"The p r o b l e m c o n s i s t s
[formulas]..."
15th p r o b l e m ,
see K l e i m a n
in this:
(For some [1976].)
Yet
limits of v a l i d i t y of the g e n e r a l i z e d de
it w o u l d be b e t t e r to r e f i n e the
is v a l i d w h e n e v e r the
15th
and w i t h an e x a c t d e t e r m i n a t i o n of the limits
r a t h e r t h a n d e t e r m i n i n g the Jonqui~res
signifi-
in the s t a t e m e n t of his
of t h e i r v a l i d i t y those g e o m e t r i c a l n u m b e r s information
line as a c o m p o n e n t
sort may be w h a t H i l b e r t had in m i n d w h e n he u s e d
to e s t a b l i s h r i g o r o u s l y
more
f o r m u l a y i e l d s via
the c o n d i t i o n s m a y admit an i n f i n i t e
"limits of t h e i r v a l i d i t y " ,
The p r o b l e m
enumerative
left o p e n to do.
curves that have an m - f o l d
in some cases,
n u m b e r of s o l u t i o n s ,
problem.
is m u c h
For
a f o r m u l a in the case of s e v e r a l g i v e n curves.
w i l l be c o u n t e d as m a k i n g Thus,
there
Moreover,
s i g n i f i c a n c e of the formulas.
long ago that de J o n q u i ~ r e s ' s
symbolic multiplication
the phrase,
f o r m u l a was given
for h i g h e r d i m e n s i o n a l v a r i e t i e s .
s t u d i e d the e n u m e r a t i v e
It was o b s e r v e d
cance.
a g i v e n n u m b e r of
[1981], who t h e n w e n t on u s i n g the same m e t h o d s to o b t a i n
formulas
g i v e n curves.
for
and that pass t h r o u g h an
A r e c e n t p r o o f of the
higher dimensional varieties,
However,
famous,
g i v e n c u r v e s are
o p e n p r o b l e m to find the r i g h t
f o r m u l a so that it
in g e n e r a l p o s i t i o n .
It is an
formula.
There can be an i n f i n i t e n u m b e r of i r r e d u c i b l e curves e a c h m a k i n g a s i n g l e c o n t a c t of g i v e n o r d e r w i t h a s i n g l e g i v e n curve, finite n u m b e r is finite.
although a
is e x p e c t e d and a l t h o u g h the n u m b e r of p o i n t s of c o n t a c t
This p o s s i b i l i t y was p o i n t e d out by Hefez
p r o v i d e d the f o l l o w i n g example,
which
(pvt. com.), w h o
is v a l i d in any c h a r a c t e r i s t i c :
185
in t h e
2-parameter
Y = x + x 7 + y8 in f a c t
fact
c a n be p r o v e d
apply
finiteness
Thm
instance, power
given
curves
is o p e n
curve
characteristic.
where
formula > 3
is j u s t t h e
of Laksov
[1981]
it y i e l d s
multiplicity formulas
coincide
is r e f l e x i v e , from the
curve
and
basis
of Laksov's
the matter When curves
method
of several
is y e t m o r e
if not,
PiHcker in a n y
tangent
the with
formula.
where
of the tangent m = 2 . m
is t h e
(possibly
q+1 .
conics
has been
analyzed
work,
and this
case
make
and m
is a
with
q3+I de
a contact n
the
is
9
intersection The
if t h e
degree
(pvt.
Thm.
present
curve.
case
of
3n(n-2),
hand,
m = 2
in t h i s
by Hefez
Now,
is t h e
inseparable
is m o r e
For
intersects
in t h e
Moreover,
m = 2
q
point
On the other
characteristic,
if
involved.
at e a c h o f t h e
multiplicity
n(n+(n-3)m),
then
This
to
contacts
where
at a g e n e r a l
q ; the with
+ yq ,
of degree
if a n d o n l y
of c o n t a c t .
establishes
curve
to its d u a l
c a s e of v a r i a b l e
issue
The tangent
point
in c h a r a c t e r i s t i c
it s u f f i c e
a smooth plane
at a g e n e r a l
no e n u m e r a t i v e
and developed
that
formula
this,
(1876)
lines
is v a l i d
like
conic
by Vainsencher,
of
familiar
the
case
every
used
in fact,
y = x q+1
for t h e n u m b e r
with
by Halphen
The
curve
with multiplicity 2 xq = x intersects
multiplicity
case
smooth
at l e a s t
of points
[1981];
the
there
Jonqui~res's
which
the
holds
curve
for i n v e s t i g a t i o n .
characteristic,
consider
that,
the Wronskian
of points.
of the
points
9. M o r e o v e r ,
(smooth)
In a c a s e
of the method
introduced
form by Laksov
7.
formula
number
a variant
number
In p o s i t i v e
least
out
the
3 at t h e o r i g i n ,
original
of the
one or more
at
(weighted)
method
abstract
Laksov's
the
intersect
least
it t u r n s
by using
the Wronskian
in a g e n e r a l
at
de J o n q u i ~ r e s ' s
yields
that
multiplicity
Nevertheless,
formula
namely,
the
with
seem that
significance. the
of c o n i c s
with multiplicity
intersects
it w o u l d
0 ,
family
curve
of the map
also). com.)
complicated.
two
The
on the
Beyond
that,
is o p e n . the plane
of degree
r
is e m b e d d e d become
the
via
the
r-fold
hyperplane
Veronese
sections;
map,
hence,
then as w a s
the ob-
186
served
long ago,
de J o n q u i ~ r e s ' s
theory of e n u m e r a t i o n 130 years, Success,
Vainsencher (i) linear
[1981] spaces
§8),
involved
success
of a r b i t r a r y especially
different.
special
they have
sort among
introduced
and d e v e l o p e d
[1886]).
], Ran
ingenius number
in this
these:
lines
of a r b i t r a r y
dimension
Two special
cases
of t r a n s c e n d e n t a l equations.
points
Such v a r i e t i e s
were
are d e f i n e d
named p a n a l g e b r a i c
X.
The g e n e r a l i z a t i o n
polynomial
I
hand,
line,
an idea [1879]
includes
[1983], open
Laksov are
and linear
spaces
I
(algebraic)
is d e f i n e d
[1901]. family
as follows.
w i t h the p r o j e c t i v i z a t i o n
Then,
differential
by L o r i a
is e f f e c t e d
to families
in the n o t a t i o n in i n h o m o g e n i o u s
The of
Identify
of the co-
of Section
2, the
coordinates
by a
equation
(4.5)
and this
of
have
enough,
(1877;
by a l g e b r a i c
curves
Pr1(F1)
of co-
the c o i n c i d e n c e s
been g e n e r a l i z e d
plane
subset
Colley
dimension,
is the case of a l-parameter
of the plane.
naturally
The next cases
case
bundle
varieties
lines
in the two cases
[1980],
special
tangent
(ii)
surfaces.
that
correspondence
and
(ii), on the other
first
the incidence
curves
work on both cases
of the t h e o r e m have
varieties
in two cases,
by S c h u b e r t
[1981].
isolated (see
on a v a r i a b l e
of a r b i t r a r y
cutting
~3
idea of e n u m e r a t i n g
and Harris
varieties
in
special
In case
recent
and V a i n s e n c h e r
cutting
cutting
connection
Some of the most
[1982],
from the
only
(i) , they have,
of curves.
a given
Aside
employed
For over
a lot of attention.
has been a c h i e v e d dimension
as part of the
spaces.
to surfaces
hypersurfaces,
Arbarello-Cornalba-Griffiths[198
attracted
planes
In case
been based on the
linear
limited.
The m e t h o d s
properties
of a g i v e n
§§33,4;
somewhat
2, and surfaces.
been b a s i c a l l y
secant
continually
and t r i t a n g e n t
varieties,
dimension
has
has been
case of b i t a n g e n t
cutting
of e x c e p t i o n a
this t h e o r y
however,
formula may be v i e w e d
P(x,y,dy/dx)
equation
is s a t i s f i e d
= 0 ,
by almost
all
X.
Now,
consider
a plane
187
curve
V
in g e n e r a l
pression formula tion the
for t h e to
that
expression
various
number
(2.22),
is n o t
of
It is a n o p e n
of the
bundle
which
been
the projection
in t h e d e r i v a is.
families
- including
cycloids,
parameter
Thus,
evolutes,
generalized
(see Z e u t h e n - P i e r i
several
ex-
Pr1(F1)
curves,
has
differential
p:
CV
equation
(= CN_IV)
irreducible
is t h e
[1915]
families
case
n ° 34).
and higher
I ~ Y ,
Y .
contact
the
accounts
for t h e b e -
of a subvariety
variety
canonical
It is not
of
I
is s m o o t h
D
0),
More
structure
structure
defined
V
of
precisely,
the
on the projectiv by the tautolog-
degeneration
by the
criterion y = pN
of
the
then
D
analogous
if
open
lies
and
in t h e
sense
in
now
~ = 0
and
of
D
Part
or
it m a y
recalled
is t h e
in S e c t i o n
"semi-l-form"
on
is a s u b -
D = CpD
of the
the
a
case may
by replacing Segre-Wallace
I, f o l l o w s . I
if a n d
under
general
C.
so in
Proposition, C V n
directly
Similarly,
D
vanishes
(it is a l w a y s
0
be p r o v e d
w
if t h e r e s t r i c t i o n ,
(a) o f t h e
n = N-I ;
to p r o v e
that if
CpD ; whence,
N-I .
for
it e a s y
conversely,
subset
semi-(N-n)-form.
w*
p.355)
in c h a r a c t e r i s t i c
follows
for r e f l e x i v e n e s s , and
that,
equation
dimension
case,
Ex.,
w = 0
to p r o v e
the behavior
from this
[1978]
on a dense
V ,
,I I ~ p ~y ~ ~I "
equation
hard
is o f p u r e
describes
be d e r i v e d
the
satisfying
characteristic if
0i(-I)
(see A r n o l d
satisfies
D ~ pD ,
if
case
by the
w:
variety
w
that
of the
"semi-l-form"
I ~CV "
which
but
matters
for p a n a l g e b r a i c
variety
smooth,
is g o v e r n e d
CV
only
On a p p l y i n g
that what
trigonametric
to t r e a t
conormal
It is w e l l - k n o w n
in
V.
algebraic
of s u r f a c e s
partial
(4.6)
that
evident
the derivation
varieties.
cotangent
ical
consider
touching
as w e l l
second
problem
an N - d i m e n s i o n a l ,
ized
are
family
An algebraic
behavior
X
spirals,
The
of a l - p a r a m e t e r
havior
X
is v a l i d
etc.
dimensional
of
and
it b e c o m e s
the
families
catenaries,
position,
viewed
Indeed,
as t h e p r o -
188
jectivization hence,
CV
use more Perhaps
of the cotangent
satisfies
bundle
~* = 0 .
~N
,
.etric t h e o r y
theory
then
clearly
It is an o p e n p r o b l e m
of the differential-g£ in t h i s w a y t h e
of
of duality
~+~*
= 0 ;
to a l g e b r i z e
of contact
and
manifolds.
c a n be e x t e n d e d
to n o n e m b e d d e d
varieties. Clebsch
(1873)
differential However,
was
equations
Fouret
the and
(1874-8)
of p a n a l g e b r a i c
curves
and
notes.
surfaces
this
for r e c o g n i z i n g
are a l g e b r a i c a l l y
Zeuthen
preparing
The proof
above
C.
originally
one of
is c a r r i e d
out
of this work understood
equation
[1982],
different
tack,
CW .
infinitely given
and
q > 1 ;
W
curves and
CV ~ W here,
There
can have
problem
are equal CV
is a n o t h e r
the
~*
open problem
V
to e x p l o r e
§7.3)
the dual
separable
theory.
[1980],
the but
it
All
little
restrictions
is a s u b j e c t
but
whose
will
curve
theory this
issue.
be r e q u i r e d . to t h o s e
and
On a of Indeed,
of the
h o w to c o n s t r u c t V*
is e q u a l
inseparable
s
that
summarized
the behavior
showed
and
integers
= 0
of
in a d d i t i o n
([1956]
to g i v e n
satisfies
of the
qualitative
to u n d e r s t a n d
solutions
such that
such that
n ° 34.)
important,
complex-analytic
A new approach
Wallace V
equa-
and usefulness.
of a very
a degeneration
of the
characteristic.
For e x a m p l e ,
curve
the map
to p a r t s
on
its s o l u t i o n s .
it is an o p e n p r o b l e m
~ = 0
many
on
based
[1915]
in K l e i m a n
is to see w h a t
articles
is e s s e n t i a l l y
interest
a solution
theory
differential
expositon
criterion
independent
under
it is an o p e n
in p o s i t i v e
the equation
puts
a method
Zeuthen-Pieri
an u p - t o - d a t e
and that
C V n
to be r e l a t e d
by Merle
form
of
of
correspondence.
of B.S.M.F.
certain
as p r e s e n t e d
toward
open problem,
The behavior
CnV
(1910),
contributes
(See
between
the enumerative
proposed
or n o t
Segre-Wallace
in 3 s t e p s
a differential
appears
Segre
incidence
founded
(1880)
integrable.
_is c u r r e n t l y
a connection
in a s e r i e s
whether
Ferrarese
of the
of the
independently
Rendus
tions
to r e c o g n i z e
subvarieties
and Comptes theory
first
q
to a
degrees
provided
of
only
CV % CW . solution
would
mean
a direct
189
refinement
of the theorem.
of the
varieties
that
p
the v a r i e t i e s
It is to d e t e r m i n e
VI,O-.,V p X
what
was
I (or, more
and the r e m a r k
suggested
earlier)
below
is what
in the
statement
of his
number
of these
X , be e s t a b l i s h e d
limits
of their validity".
should be d e t e r m i n e d
sitions
of
VI,-..,V p
the
5-parameter
smooth conics. points join,
in
~2
If no two
such that each and
if there
of t h e m or passes touching
V. l
conics,
passes
through
is no pair of lines through
their
V I , . - . , V 5 , each
X
This result was p r o v e d ized to curves of the total
number,
is s m o o t h
it is assumed
that
their
The case
degree.
ably the next case Associated fying v a r i a b l e reflects
almost
extension;
all
in w h i c h
X
po-
vary
Vl,...,V 5
in
be
is no pair of
Vi
then there
and appears
their
touches are
3264
one X's
with multiplicity,
N Vi))-
and all
to an e n u m e r a t i o n algebro-geometric
V. 1
([1977]
§7).
an e x p l i c i t
V. l
It was g e n e r a l determination
[1983].
are general
are given
However, there curves
arbitrarily
of a l g e b r o - g e o m e t r i c conditions,
of the situation,
it. The group may be v i e w e d namely,
let
such that each
(without
the
2,
if there
in H e f e z - S a c c h i e r o X
of a p p e a r a n c e
of
is p r o b -
to consider.
the c o m p l e x i t y
determining
degree
of course)
of the
one of t h e m or touches
in F u l t o n - M a c P h e r s o n
of a r b i t r a r y
determination
and let
intersection,
5 zi=1(4 - c a r d ( X
(4.7)
he asked
w h e n the r e l a t i v e
but
t o u c h each other,
i
(instead of
like the total
the m u l t i p l i c i t i e s
generally
them
degenerate.
(reduced) V
this
so
see Part
had in mind when
in any c h a r a c t e r i s t i c
family of
mcq ,
Possibly
"with an exact
more
are m i l d l y
family that t o u c h
that numbers,
At any rate,
X's
it).
positions
and s u f f i c i e n t
generally,
Hilbert
15th p r o b l e m
of the
For example,
are n e c e s s a r y
in the given p - p a r a m e t e r
are counted w i t h m u l t i p l i c i t y (d) of the T h e o r e m
that
the r e l a t i v e
it is the e x t e n s i o n
there
and there
is a group,
satiswhich
is the p r o b l e m of
as the Galois of the
figures
group of a field
field of d e f i n i t i o n
of
190
the g e n e r i c
conditions
fying them.
Alternatively,
namely, paths
yields
the group may be v i e w e d
a g r o u p of p e r m u t a t i o n s of the
l i s h e d by H a r r i s numbers.
The e q u i v a l e n c e
[1979].
and in m i x e d
Harris
[1979]
of d e g r e e
2N-3
cases of the
in
they are n o n t r i v i a l .
and the
d ,
there
found
in the
figures
are a r b i t r a r y
and of
19th century, the g r o u p
and
is the
a transposition.
coalesce.
A
In the case of the
of a t r a n s p o s i t i o n
(4.7) of F u l t o n - M a c P h e r s o n .
is t h e r e f o r e Similarly,
p r o v e d that the g r o u p is the full s y m m e t r i c g r o u p
in the case of c u r v e s of d e g r e e of v a r i o u s
In the
is a s t r a t u m of p a r a m e t e r v a l u e s w h e r e
the e x i s t e n c e
a c o n s e q u e n c e of the e x p r e s s i o n
V. 1
5 conics.
H a r r i s p r o v e d this r e s u l t by s h o w i n g that the
5 conics,
[1983]
flexes and
of the b i t a n g e n t s of a quartic,
e x a c t l y two of the c o r r e s p o n d i n g
Hefez-Sacchiero
the
the lines on a h y p e r s u r f a c e
is d o u b l y t r a n s i t i v e and that it c o n t a i n s
touching
in p o s i t i v e char-
in these cases:
In each of the r e m a i n i n g cases,
s y m m e t r i c group.
curves
s a t i s f y i n g the
3264 conics t o u c h i n g
the g r o u p s w e r e
t r a n s p o s i t i o n exists b e c a u s e
Vi
figures
characteristics.
flexes of a cubic,
the lines on a cubic,
conics
along various closed
H a r r i s w o r k e d e x c l u s i v e l y o v e r the c o m p l e x
curve of d e g r e e ~N,
satis-
of the two v i e w p o i n t s was e s t a b -
d e t e r m i n e d the group
b i t a n g e n t s of a p l a n e
figures
as a m o n o d r o m y group;
It is an o p e n p r o b l e m to d e v e l o p the t h e o r y
acteristic
group
field of d e f i n i t i o n of the
v a r y i n g the parameters of the c o n d i t i o n s
initial conditions.
full
by the
degrees
(reduced)
m ~ 2
touching
> 2 . Once again,
plane
curves
p =
general
in w h i c h the
is p r o b a b l y the next
case to
consider. Chasles describe
felt that the c h a r a c t e r i s t i c
it for e n u m e r a t i v e
over 200 examples. diameter,
Most were
angle b i s e c t o r ,
purposes. stated
etc.
numbers of a family s u f f i c e to
He s u p p o r t e d this c o n t e n t i o n w i t h
in a f f i n e
However,
and m e t r i c
they can be r e p h r a s e d
of a line at infinity,
a pair of c i r c u l a r points,
dramatic,
is e a s i l y u n m a s k e d .
but t a n g e n c y
terms
etc.
like focus, in terms
The e f f e c t
On the other hand,
is
Halphen
191
(1878)
found some
conics
in a family
that
linear
combination
of the c h a r a c t e r i s t i c
situation work up
can be justified
loci
satisfy
For v a r i e t i e s open p r o b l e m
one of them
of h i g h e r
n-folds
degree,
little
to find a c o m p l e t e
(de Jonqui~res) number
condition
of c o n t a c t s
at from Schubert's
point
express
the c o n d i t i o n
ditions,
independent
as the n u m b e r
of g i v e n order with of v i e w
of curves
V , and to express in c e r t a i n
standard
sinto
in any c h a r a c t e r i s t i c . For example,
a given 3),
combination
blowing-
the theory
numbers
in a family
(see Section
the
Halphen's
of the a p p r o p r i a t e
pN
curves
to a
rectified
of e q u i v a r i a n t
is known.
on the plane
equal
numbers.
set of c h a r a c t e r i s t i c
as a linear of
simply
to g e n e r a l i z e
in
of smooth
Halphen
theory
analysis
It remains
of q u a d r i c
is not
numbers.
Hironaka's
com.).
The number
characteristic
or by a direct
pvt.
families
on conics.
additional
by using
[1980]
(Casas,
p-parameter
given
conditions
by i n t r o d u c i n g
(see K l e i m a n
gular
subtle
for the
to make
curve
a
V . Looked
the p r o b l e m
of c e r t a i n
it is an
is to
standard
con-
the c o m b i n i n g
coefficients
families
satisfy
that
the
condition. The Theorem,
howsoever
more than a first like the three still
step in the rigorous
famous
necessary
it is refined
numbers
verification
mentioned
themselves,
finding
is by virtue
of
cation.
For example,
the c h a r a c t e r i s t i c
ily of all conics of conics
(1.12)
in the plane
touching
5 others
also
are
numbers
sufficient
For the tic numbers
were
I, 2, 4, 4, 2,
It is
families.
For
characteristic the verifi-
5-parameter
I; hence,
3
+
fam-
the number
\I/
to find the c h a r a c t e r i s t i c
of all conics
found
of the
numbers,
is
it is s e l d o m easy family
of the
to c o m p l e t e
numbers
\2/ Nevertheless,
of e x p l i c i t
the c o r r e s p o n d i n g
numbers
can never be
at the very beginning.
to find the c h a r a c t e r i s t i c
the three numbers
and generalized,
in the plane,
by the ancient
Greeks,
numbers.
some of the c h a r a c t e r i s -
and the rest,
by N e w t o n
192
(Principia,
1687).
Zeuthen
(1865),
inspired by C r e m o n a
a method for finding the c h a r a c t e r i s t i c of conics.
(1864), d e v e l o p e d
numbers of a r b i t r a r y families
The key idea is to pass via b a s i s - c h a n g e relations between
the basis of the two e l e m e n t a r y conditions,
to pass through a point and
to touch a line, and the basis of the two d e g e n e r a c y conditions, a line-pair and to be a d o u b l e - l i n e eralized,
(see Kleiman
[1980]).
to be
S u i t a b l y gen-
the m e t h o d has been used in other cases as well.
The charac-
teristic numbers were found for the family of all conics in space by Chasles
(1865), and for that of all q u a d r i c s
Schubert
in space by Zeuthen
(1866).
(1894) d e v e l o p e d an a l g o r i t h m for finding the numbers for the
family of all q u a d r i c n-folds
in
~N.
Recently,
these numbers and the
general a l g o r i t h m were e s t a b l i s h e d r i g o r o u s l y in i n d e p e n d e n t works, each with its own v a l u a b l e point of view, Vainsencher Thus, 666,
[1982], De C o n c i n i - P r o c e s i
in particular, 841,
by van der W a e r d e n
(pvt.ms.,
[1982], and L a k s o v
1981),
(pvt.ms., 1982).
the r i g o r o u s v e r i f i c a t i o n of the numbers 3264 and
088 is complete.
Maillard
(1871)
and Zeuthen
(1872)
independently
found the charac-
t e r i s t i c numbers of the 7 - p a r a m e t e r family of c u s p i d a l plane cubics, then those of the 8 - p a r a m e t e r family of nodal cubics, of the 9-parameter family of all plane cubics.
then finally those
Schubert
(1874, 5) refined
this w o r k and went on to find the c h a r a c t e r i s t i c numbers of the 12-parameter family of all twisted cubic space curves.
Zeuthen
(1873)
found the
c h a r a c t e r i s t i c numbers of the 14-parameter family of all plane q u a r t i c s via a lengthy s t e p - b y - s t e p d e t e r m i n a t i o n of the numbers of variou.s subfamilies of singular quartics. see Schubert Sterz
[1879].)
Recently,
[1982], and StrUmmer
Ellingsrud
[1982], and P i e n e - S c h l e s s i n g e r
of t w i s t e d cubics. 783,
Sacchiero
(pvt. com.)
e n u m e r a t i o n of plane cubics. Piene
(For a d i s c u s s i o n of all of this work, (pvt. com.),
Speiser
i n d e p e n d e n t l y have begun the (pvt. com.), Harris
(pvt. com.),
[1982] have begun the e n u m e r a t i o n
Perhaps, before too long, the number 5, 819,
680 will be verified.
(pvt. com.),
539,
193
The 19th century w o r k on finding the c h a r a c t e r i s t i c numbers of families of curves of higher degree is rich and lovely.
Understanding
it well enough to v i n d i c a t e
it and to c o n t i n u e it, is p o s s i b l y the most
important part of H i l b e r t ' s
15th p r o b l e m
r e m a i n i n g open.
alone has the full stature of a Hilbert problem. wrote,
This part
It is, as Hilbert
"so clear that you can explain it to the first man you meet on
the street"
and yet "from the d i s c u s s i o n of
[it] an a d v a n c e m e n t of
science may be expected."
References
Arbarello,
E.-Cornalba,
M.-Griffiths,
P.-Harris,J.
[1980]: Topics in the
theory of al~ebraic curves, to appear in the P r i n c e t o n Math. Arnold, V.
[1978]: M a t h e m a t i c a l methods of c l a s s i c a l mechanics,
lated by K. V o g t m a n n and A. Weinstein, Springer, Colley,
S.
Cremona,
60,
(1978).
thesis, Mass.
J.
York
trans-
G r a d u a t e Texts in Math.
[1983]: On the e n u m e r a t i v e g e o m e t r y of s t a t i o n a r y m u l t i p l e -
points, Coolidge,
New York
Series.
Institute of Tech., Cambridge,
Mass.
(1983).
[1959]: A t r e a t i s e on a l g e b r a i c plane curves, Dover, New
(1959). L.
[1862]:
I n t r o d u z i o n e ad una teoria g e o m e t r i c a delle curve
piane,
Bologna
(1862)
Curven, De Concini,
t r a n s l a t e d by M. Curtze, C.-Procesi,
print, Univ. Demazure, M.
[1974]:
C.
Fulton,
35
[1982]: C o m p l e t e symmetric varieties,
Scient.
Ec. Norm.
R.
[1977]:
[1905]:
Sup.
(4) 7(1974),
53-88.
"Defining algebraic intersections",
(Proc. Sympos.,
1977) Lecture Notes in Math.,
Univ.
Troms¢,
687. Springer,
Troms¢ Norway,
Berlin
(1978),
1-30.
"La teoria delle formule d ' i n c i d e n c e e di p o s i t i o n e
speciale e le forme binarie", 1041-62.
pre-
(1982).
(1934), 396-433.
W.-MacPherson,
G.
(1865).
"Sur la t o p o l o g i e de certains espaces homog~nes",
Algebraic Geometry
Giambelli,
Greisswald
" D ~ s i n g u l a r i s a t i o n des vari~t~s de Schubert Ann.
[1934]:
Ann. of Math.
C.
di Roma
g~n~ralis~es", Ehresmann,
= E i n l e i t u n g in einer Theorie der ebenen
Atti Acc.
Scienze Torino XL
(1904-5),
194
Grayson,
D.
Com. Harris,
[1979]:
"Coincidence formulas
in A l g e b r a 7 (16) (1979),
J.
[1979]:
J. 46
in e n u m e r a t i v e geometry",
1685-1711.
"Galois groups of e n u m e r a t i v e problems",
Duke Math.
(1979), 685-724.
Hefez, A.-Kleiman,
S.
[1983]:
"Notes on duality for p r o j e c t i v e
varieties",
to appear. Hefez, A . - S a c c h i e r o ,
G.
[1983]:
lem for plane curves", Hodge, W.-Pedoe,
D.
[1973]:
to appear.
[1952]: Methods of algebraic geometry,
C a m b r i d g e Univ. Katz, N.
Press.
(1952), r e p r i n t e d
"Pinceaux de Lefschetz:
Lecture Notes in Math. Kempf, G.-Laksov, calculus", Kleiman,
S.
D.
[1974]:
S.
[1976]:
28
S.
(1974), 287-297.
Mathematical Developments
Proc. Sympos.
in Pure Math.,
S.
Sijthoff and N o o r d h o f f [1980]:
introduction",
S.
[1981]:
[1981]:
on curves", in Ann. le Barz, P.
Loria, G.
(1980),
Birkha~ser
MAA Studies
in Math.
Proc.
18th Scandina-
(1981).
"Wronskians and PiHcker formulas
Nice
Oslo 1976, P. Holm
117-138.
Institut M i t t a g - L e f f l e r
Report No.
for linear systems 11
(1981), to appear
Sup.
[1982]: Quelques
[1901]:
Sympos.
"Concerning the dual variety",
Ecole Norm.
preprint,
Proc.
"Chasles's e n u m e r a t i v e theory of conics. A historical
vian Congress of Math., Laksov, D.
XXVIII, A.M.S.,
(1977).
Studies in A l g e b r a i c Geometry,
20, A. S e i d e n b e r g editor Kleiman,
arising from
"The e n u m e r a t i v e t h e o r y of s i n g u l a r i t i e s of mappings",
Real and complex singularities,
Kleiman,
formula of Schubert
153-162.
(1976).
[1977]:
editor,
SGA71I,
(1973), 212-253.
"Problem 15. Rigorous f o u n d a t i o n of Schubert's
Hilbert Problems,
Kleiman,
Berlin
"The t r a n s v e r s a l i t y of a generic translate",
e n u m e r a t i v e calculus",
Providence
(1974),
II,
1968.
"The d e t e r m i n a n t a l
132
vol.
th~or~me d'existence",
340, Springer,
[1974]:
Acta Math.
C o m p o s i t i o Math. Kleiman,
"The Galois group of the t a n g e n c y prob-
formules m u l t i - s ~ c a n t e s pour les surfaces,
(1982).
"Le curve p a n a l g e b r i c h e " ,
delle Science di Praga
Memorie della R. Societ~
(1901), r i s t a m p a t a con correzioni ed aggiunte
nel Le Mat. pure ed a p p l i c a t e
II, Num.
4-5
(1902),
1-24.
195
MacPherson,
R.
[1974]:
Ann. of Math. Merle, M.
[1982]:
100
"Chern classes
for singular algebraic varieties",
(1974), 423-32.
"Vari~t~s polaires,
s t r a t i f i c a t i o n s de W h i t n e y et
classes de Chern des espaces analytiques S~m. Bourbaki, Piene,
R.
[1978]:
Ec. Norm. Piene,
R.
Nov.
complexes
11
R.-Schlessinger,
M.
BirkhaHser,
I. [1971]:
Proc.
Boston
Enumerative
Sypos. Nice
(1980),
(1982), 37-50.
[1983], On the Hilbert scheme c o m p a c t i f i c a t i o n
of the space of twisted cubics, preprint,
sympos.
Scient.
"Degenerations of complete twisted cubics",
le Barz and Hervier editors,
Porteous,
Ann.
(1978), 247-76.
g e o m e t r y and c l a s s i c a l algebraic geometry.
Piene,
[d'apr~s L~-Teissier],
600.
"Polar classes of singular varieties",
Sup.
[1982]:
1982, exp.
Univ. of Oslo
"Simple s i n g u l a r i t i e s of maps",
I, Lecture Notes in Math.
(1983).
Liverpool
192, Springer,
Berlin
singularity
(1971),
286-307. Ran,
Z.
[1982]: The class of a Hilbert scheme inside another, w i t h
a p p l i c a t i o n s to p r o j e c t i v e g e o m e t r y and special divisors, ment, Math. Dept. B r a n d e i s Univ. Roberts, J.-Speiser, triangles
R.
[1980]:
"Schubert's e n u m e r a t i v e g e o m e t r y of
from a m o d e r n viewpoint",
C o n f e r e n c e at Chicago Circle, Berlin Schubert,
H.
announce-
(1982).
A l g e b r a i c Geometry,
Lecture Notes in Math.
Proc. of a
862, Springer,
(1981). [1879]: Kalk~l der a b z ~ h l e n d e n Geometrie,
Teubner,
Leipzig
(1879), reprinted w i t h an i n t r o d u c t i o n by S. Kleiman and a list of publications Schubert,
H.
a s s e m b l e d by W. Burau,
[1886]:
Springer,
H.
[1903]:
26
(1886), 26-51.
"Gleichungen z w i s c h e m B e d i n g u n g e n bei special Lage
linearer R~ume", Mitt. math. Ges. H a m b u r g 4 (1903), Scott,
(1979).
"Die n - d i m e n s i o n a l V e r a l l g e r m e i n e r u n g e n der funda-
m e n t a l e n A n z a h l e n unseres Raums", Math. Ann. Schubert,
Berlin
D.
[1964]:
"Andrew Zobel", J. London Math.
Segre, C.
[1892]:
"Intorno alla storia del p r i n c i p i o de c o r r e s p o n d e n z a
e dei sistemi di curve",
Bibl. math.,
Soc.
104. 39
(1964
, 566-7.
6 (1892), 33-47 = Opere, vol.
I, 185-197. Segre, C.
[1912]:
" M e h r d i m e n s i o n a l e RaHme",
Wissenschaften,
Teubner,
Leipzig
E n c y c l o p a d i e der M a t e m a t i s c h e n
(1912-34)
III, 2, 2, C7,
669-972.
196
Sterz, U.
[1982]:
Ordnung", Vainsencher, 36
"Beruhrungsvervollst~ndigung
Beitr. Alg. Geom.,
I. [1978]:
(1978),
Vainsencher,
"Conics in characteristic
2", Compositio Math.
101-12.
I. [1981]:
Trans. A.M.S. Vainsencher,
f0r ebene Kurven dritter
in press.
267
"Counting divisors with prescribed singularities", (1981), 399-422.
I. [1982]:
"Schubert calculus for complete quadrics",
Enumerative geometry and classical algebraic geometry. Nice
(1980), le Barz and Hervier editors,
BirkhaHser,
Proc. Sypos. Boston
(1982),
199-235. Urabe, T. [1981]:
"Duality of numerical characters of polar loci", Publ.
Res. Inst. Math. Wallace, A.
[1956]:
Sci.
17 (1981), 331-345.
"Tangency and duality over arbitrary fields",
Proc.
London. Math. Soc. 3 (1956), 321-342. Zeuthen, H.-Pieri, M.: math~matique, Zobel, A.
[1958]:
"G~om~trie ~num~rative",
Teubner,
Leipzig
Encyclop~die des science
(1915), III, 2, 260-331.
"On the contacts between the varieties of two systems",
Rend. di Mat.,
17 (1958), 415-422.
UN P R O B L E M E POUR
LES
DU TYPE
BRILL-NOETHER
FIBRES
VECTORIELS
Franco
Soit
X une
g, d ~ f i n i e
courbe
sur
un
compl~te,
corps
un f a i s c e a u
gr~
s. D a n s
ensemble
o~
ferm~
de P i c ~ :
W~(F)
=
r et m sont Dans
sique. alors
le cas
En
effet
note
dim
entiers
Nest
genre
libre
on veut
H°(X,F~L)
F sur
X de d e -
~tudier
le s o u s -
~ r+1}
fixes.
s = I on a i e si
de
c l o s de c a r a c t @ r i q u e
localement
cette
{i 6 P i c ~ :
deux
lisse,irr~ductible
k alg~briquement
z~ro. C o n s i d ~ r o n s d e t de r a n g
Ghione
un
probl~me
faisceau
de B r i l l - N o e t h e r
inversible
clas
de degr~
n,
l'isomorphisme m+n ~ ~Ic x .
Pic~ d~fini
par multiplication
avec
N,
donne
pour
tous
m @ 2,
l'i-
somorphisme W~(F~N)
~ Wr (F) m+n
-
et
donc,
sis
W ~(0X ) , que des
s@ries
r. P o u r
= I, on p e u t nous
~crivons
lin@aires
W~ on sait
Recherche
support@e
supposer
F = 0 X. E n
simplement
r a v e c W m,
s u r X de d e g r ~
que par
([4], le
met
ce cas, est
alors,
l'ensemble
de d i m e n s i o n
au moins
[3])
"Ministero
della
Pubblica
Istruzione".
198
(a)
dim
W r > T = gm
en
plus,
si
dans
la
formule
(a)
[W~] o~
[W~]
d6note
Wrm d a n s teta,
la c l a s s e
l'anneau
un n o m b r e
E ~
de
donn6
alors:
• @g-T
Pic~,
@quivalence
entier
il y a @ g a l i t 6 ,
fondamentale
de C h o w
~ signifie
(r + I) ( r - m + g )
--
du
@ est
num@rique
sous
ensemble
la c l a s s e des
du
cycles,
f@rm@ diviseur
et
e est
par r
i.' = I I (r-m+g+i) ' i=O (b)
Si
X est
les,
alors
dans
Dans (a)
en
face
@tudiant
Dans
p = sm+d@galit@,
dans
sa
vari~t@
des
modu-
a 6galit@.
s = 2 on
peut
s@ries
note
dim
o~
g@n6rique
on
obtenir
lin6aires
P(F) , unisecants
cette
y-a
on
les
localement
(a)
courbe
(a)
le cas
r~gl@e
sceau
une
veut
les
g~n~ral,
wr(F) m
> T(F)
(g-l) (s-l).
En
des
r6sultat diviseurs
g~n~ratrices
d@montrer
libre
un
alors
= g-
que, on
pareil de
la
sur
[I]. si
F est
un
fai-
a
(r + 1) ( r - p + g )
plus,
si
dans
la
formule
(a)
alors
[Wmr(F)]
_ ~(F)sg-<
F) . @r-<(F)
o~ r
(F)
L'id@e
de
=
i [ i=O
la d ~ m o n s t r a t i o n
l! (r-p+g+i)! est
celle
de
consid@rer
le
il-
199
schema
Q u o t ~ qui
rang s - I suite
param~trise
et de degr~
d +m.
les q u o t i e n t s Si G est
de
F coh~rents
un tel q u o t i e n t
de
on a la
exacte: 0 ~ I(G)
-* F -+ G -~ 0
et H o m ( I ( G ) , 0 X) = I(G) v est un f a i s c e a u m. On a alors
un m o r p h i s m e ~m:
d~fini,
avec
abus
cas s = I Q u o t ~ consid~r~
dans
consider~e
en
En tous
Quot~-~
le cas
Pic X
par ~m(G)
= H i l b ~ = sm(x) Dans
de degr~
fonctoriel
de n o t a t i o n s ,
~4].
inversible
= I(G) v. Dans
et le m o r p h i s m e
~m est
s = 2 on a la m ~ m e
le celui
situation
[I]. cas
la fibre
du m o r p h i s m e
~m est
donn~e
par
,~-I.(L) = m ( H °(x,F~L) ~) m
et donc W~(F) Si on s u p p o s e ctible,
alors
on p e u t
=
que
{L E Pic~: le s c h e m a
appliquer
d i m ~-Im(L)
Quot~
la formule
soit
~ r}
lisse
de P o r t e o u s
et i r r ~ d u -
pour
~valuer
la d i m e n s i o n
de Wr(F) et, dans le cas de bonne d i m e n s i o n , m culer la c l a s s e f o n d a m e n t a l e du cycle Wr(F) dans l ' a n n e a u m C h o w de Pic~. M a i n t e n a n t l ' h y p o t h ~ s e que Q u o t ~ soit lisse irr~ductible [2],
est v ~ r i f i ~ e
et donc,
peut bien sultat
si on c o n n a i t
appliquer
annoyS.
ma Q u o t ~
est
si
F est
les c l a s s e s
la f o r m u l e
Donc
la p a r t i e
g~n~ral
le calcul
de C h e r n
de P o r t e o u s des
principale
dans
classes
de cette
le sens
de C h e r n note.
de et
de
de Quot~,
et d ~ m o n t r e r
cal
on
le r~
du sche-
200 m
§I. UNE D E S C R I P T I O N Dans pour
ce p a r a g r a p h e
obtenir
le s c h e m a
de d i v i s e u r s n~e
dans
dans
Gun
d~tails
On a la suite
est
On a alors
voulons
donner
une c o n s t r u c t i o n
comme une i n t e r s e c t i o n m Quot0s. La c o n s t r u c t i o n
dans
compl~te est don-
[2~.
coherent
de
F de r a n g
s-1
et de degr~
canonique
0 ~ o~ I(G)
Quot~
quotient
Quot~
nous
le s c h e m a
tousles
Soit d+m.
DU S C H E M A
I(G)
un f a i s c e a u le m o r p h i s m e
~
F ~ G ~ 0
inversible
de degr~
fonctoriel,
qui
-m.
g@n@ralise
le m o r p h i -
sme de A b e l - J a c o b i : ~m: qui
transforme
sible
le q u o t i e n t
~ Picx G dans
la classe
du f a i s c e a u
r~ L e t
soient
PI:
X ×Plc x
•
d@fini
(1)
sur X × P i c ~
PI'
P2
le f a i s c e a u
les p r o j e c t i o n s
universel
m
~ X,
P2:
un f a i s c e a u
coh@rent
de P o i n c ~
canoniques
m
Ii e x i s t e
XxPlcx
m
' Picx"
QF sur P i c ~_
([2]
par Hom
{~F,:.I) = p2 ~ ~ p ~ F ~ p ~ M )
tel que
]P (QF) = Q u o t F et
le m o r p h i s m e
(QF)
inver
I(G) ~. Consid~rons
3.10)
QU°tF
~ Pic~.
9m c o r r e s p o n d On v o i t
alors,
~ la p r o j e c t i o n imm~diatement,
canonique que
201
(2)
Y-I(L) m
= P(H°(X,F~L) ~ )
et Wr(F) d e v i e n t le s o u s - e n s e m b l e m les x E P i c ~ tels q u e d i m QF~k(x) morphisme
~ r+1.
En plus on a l'iso-
fonctoriel
(3) pour
ferm~ de P i c ~ d~fini par
W r(F~H) m chaque
~ Wr (F) - m+n
faisceau inversible
N de degr~
net
pour chaque
i n t i e r r et m. O n p e u t alors 1.14)
s u p p o s e r deg F = - d m
0 et on a alors
[2]
une s u i t e e x a c t e
(4)
0 ~
F ~
O xs
~0D
~ 0
-d o~ 0 D = i~ I 0{x~},
x i e X, x i ~ xj pour
faisceau concen~r~
sur le p o i n t x i, 0x/m i = 0{xi}.
(4) nous donne, donc les
pour chaque
j ~ jet
i, les p r o j e c t i o n s
0{xi} est le La suite
0~ ~ 0{xi} et
suites exactes
s~0{ xi } ~0 -
(5)
0 ~ Fi ~ 0 X
On a alors,
pour
chaque
f a i s c e a u c o h e r e n t M sur Pic x, la sui-
te
0
•
,
et, par la formule (6)
,
,
s
,
~ P2* aL~P10x~P2M)
p2,(~@PiFi~P2M)
~ P2*
~p~0
(I)
ix i
~ 0 ~0 Q F
I
= 11. , et cela d~fini o~ x. xi×Pic ~ xi
~0 une i m m e r s i o n
~i: ]? (QF.I) ~ ~ (Q0~)
*
{x i }~p2 M)
202
o~ le d i m i s e u r P ( q F
) =: Qi est plong@
dans P (O0s) =: Y comme
le s c h @ m a des z@ros i de la s e c t i o n de 0~0Xs ( 1 ) ~ ~ x i ) le m o r p h i s m e compos~
d~finie
par
Oy -~ ~ ( Q O ~ ] L x i ) En plus,
-+ O~Ox(1) ~*mOLx.l )"
par la formule s =
0x
(I), on d ~ d u i t que
@ ..
Q0 x
•
(s-fois)
~0x
•
et P ( Q 0
) = sm(x). Donc, si m > 2 g - 2 , ~0 s est l o c a l e m e n t fiX -X bre de rang s(m-g+1) et ses c l a s s e s de C h e r n sont d ~ d u i t e s des classes
de C h e r n de ~0 x qui sont b i e n connues.
S u p p o s o n s m a i n t e n a n t que normalis~
dans
le f a i s c e a u de P o i n c a r ~ L
le p o i n t x ° E X: =
Lx o
0p.
l'~quivalence
m
ic x
et soit Q un d i v i s e u r de Y dans On a alors
soit
la classe d ~ f i n i e par 0~0
(I).
alg~brique
Qi - Q. Les suites
(4) et
(5) nous d o n n e n t
l'isomorphisme
([2]
1.2) -d
]P(QF) et,
si F est g @ n ~ r a l dans
=
X 1D(QFi ), i=1
sa variet@
des modules,
tion des d i v i s e u r s ]P(0F ) = Qi est d~finie et ]P(0~F) est i r r @ d u c t i b ~ e On a alors
la
l'intersec-
et t r a n s v e r s a l e
(si de d i m e n s i o n positive)
[2].
[2]
203
Proposition Soit soit
0 = Q0 x le
Y = P (Q •
la c l a s s e sur
1
... O Q)
d@finie
s > 2, m
par
> 2g-2,
--
faisceau
m
s u r P i c x tel q u e ]P(Q) = s m ( x )
(s-fois)
0y (I).
et Q un diviseur
Supposons
il e x i s t e
alors
que
une
t
de Y d a n s
deg F = -d
immersion
>> 0
ferm6e
m~y : Quot F telle
que -d Quot~
o~
le Qi
lents
sont
des
diviseurs
~ Q. E n p l u s
transversale
dans
si
sa d i m e n s i o n
est
gardant
lisses
l'intersection
et
Sis
= i~I YQi
tousles
de Y a l g ~ b r i q u e m e n t des
points
diviseurs et Quot~
Qi e s t
est
~quiva d~finie
irreductible
positive.
= I la p r o p o s i t i o n d = 0, Y = ~ ( Q )
peut
et Quot~
s'enoncer
@galement
e n re-
= Y.
F
m
§2.
LES
CLASSES
DE
Soit V une corps V.
variet6
k et soit
Nous
Fun
d6notons C(t,F)
le p o l i n 6 m e
de
et
DU SCHEMA lisse
faisceau
par C(t,F) = I +c1(F)t
Segre
S(t,F) oO d = d i m V
SEGRE
est
et
QuotF
irr~ductible
localement
le p o l y n 6 m e
libre
d@finie de r a n g
de C h e r n :
+ c 2 ( F ) t 2 + ..... + C r ( F ) t r ;
d@fini
par
= I + Sl (F)t + s 2 ( F ) t 2 + ..... + S d ( F ) t d
sur
le
r sur
2~
C(t,~) Pour
les c l a s s e s
S(t,F)
de Segre o n a ~.(~r-1+i)
o0 ~: ~ ( F )
~ Vest
est la c l a s s e sceau
des
la p r o j e c t i o n
de G r o t h e n d i e c k
differentiels
t@ristiques C(t,V)
de
la f o r m u l e = si(F )
canonique,
et ~ = c1(0V(1))
F. Enfin,
de V sur k,
de V s o n t d o n n @ e s
= I.
alors
si ~V est
le fai-
les c l a s s e s
carac-
par
=: C ( t , ~ V) = 1 + c I ( V ) t + ..... + C d ( V ) t d
et d o n c C(t,V) On v e u t m a i n t e n a n t
=
calculer
V = Quot~.
Avec
les m ~ m e s
pr@c@dent,
nous
avons
Quot~
=
S(t,n v) -I S(t,a V) p o u r
notations
le d i a g r a m m e
~(2F ) c
que
dans
la v a r i ~ t ~ le p a r a g r a p h e
suivant
;
Y
= ~(2
• ~.. •
m Pie x
En u t i l i s a n t
les s u i t e s
0 ~ ~Pic~ 0 ~ ~YI
on o b t i e n t
m ~ Plc x
exactes
canoniques
" ~Y ~ ~Y !Pic~ ~ 0 (2)S~0y(-1)
~ 0y
0
2)
205
C(t,~y)
= (1-~t)s(m-g+l).c
I1-~t t
' ~ (QS)
1
et donc t , ~*Q)/(1+~t)m-g+IIS S (t,~y) = ~_S(~+~t Pour calculer
les classes de Segre de Quot F on utilise
le fait que, si F est g~n~ral, ~(QF ) est l'intersection pl~te des diviseurs Qi et donc on peut calculer mal de ~(~F) dans Y. On trouve la proposition
com-
le fibr~ no~
2
On a -d ES t ~* (~))/(l+~t)m-g+l 1 S(t,~Quot~ ) = "I I (I+~i t) (l+~t ' m i=I O~ ~ = e*~ et ~i = ~*([Qi ])" En particulier l'anneau d'~quivalence num~rique
S(t'£Quot~)
on trouve, dans
t y* (Q)) s/(1+~t) p-g+1 -= S(1+~---~ ' m
o~ p = s(m-g+1) + g - I et nous supposons
toujours
D > O.
+ d = dim Quot~
206
§3.
APPLICATIONS Pour
chaque
coefficients
A
Soit, ses,
ble
DE
a,r
s~rie
S(t)
=
dabs
un
(s(t))
Soit
p = dim G r C X, Gr =
Si
G r ~ ~ alors
ii)
Si
G r ~ ~ et
Si
Gr = 0
gradu~,
nous
posons
Sa_ I
sa
• -- S a + r _ 2
Sa-r+ 1
Sa-r+2
..-
(dans
g = dim
Y.
sens
!
de
de
>_ r}
variet~s
schemas)
Consid~rons
lisses.
r <
p.
(Parteous):
r(g-p+r)
composant
irr~ductible
de
G rest
alors (C(t,f*~)-S
(t,~ x))
alors
voulons
appliquer
ces
m: 0uot
lis
le s o u s - e n s e m -
p-g <
suivants
codimXGr
A
le
f-lf(x)
r~sultats
chaque
sa
X ~ Y un m o r p h i s m e
Ag_p+r,r(C(t,f*~y)'S(t,~ Nous
+ s o + s l t + s2 t2 + . . .
• .- S a + r _ I
g-p+r,r iii)
-1
+s_it
Sa+ I
r(g-p+r) =
-2
sa
XJdim
i)
[G r]
anneau
f:
X et
les
codimension
PORTEOUS
Laurent
fibres
{x •
avons
de
DE
=
~
f erm~
FORMULE
... + s _ 2 t
maintenant,
projectives
Nous
LA
x))
r~sultats
Pic
=
0.
au morphisme
de
207
E t a n t donn@ que ~ P i c ~ est trivial, x d@terminant r × r Ag_p+r,r(S(t,~Quot~)) En u t i l i s a n t
la p r o p o s i t i o n
Ag_p+r,r(S(t,~Quot~))
il s u f f i t
pour
de c a l c u l e r
le
p-g ~ r ~ p.
2 on a
--
_ A g _ p + r , r ( ( 1 + ~ t ) g - l - P . s ( 1 + ~ t , y*(~))S)m =
=
t ,~ ).S(~+~ t
Ag_p+r,r((1+~t)g-p(1_~
= A g - p + r , r ((1-~t)'S(t'~(~))s)m
r [ i=0
et
~(F)
=
r i=O
lemma
13]
=
(g-p+r-1+i) ! ~isP-r-T-i i! (r-i) !
oR T = g-(r+1) (r-p+g)
[ 4
--
= Ag_p+r,r((1-~t)exp(~:(8)-st)) = ~(F)-
(Q))s) =
~m(@P-r-r-1)
i! (g-p+r+i) !
En p a r t i c u l i e r
(~m)~ (~r'Ag_p+r, r (S(t,~Quot ~))) et donc, si T ~ 0, A g _ p + r , r ( S ( t , ~ Q u o t m ) Nous p ~ r r i o n s alors c o n c l u r e a v e c la Proposition Soit
- ~ ( F ) s g - T . o g -T ) ~ 0.
3
F un~ f a i s c e a u
g~n~ral
Soient nr et r deux entiers
de rang s e t
tels que
de d@gr~
d.
p = sm+d-(s-1)(g-l) > 0 et
208
0-g
< r < 0. A l o r s dim
En
particulier
si
on
a que
Wr(F) m
> T = g-(r+1)
T > 0 alors
Wr(F)
--
dim
W~(F)
=
La
spond
~
"th~or~me la p a r t i e
semble
alors
cas
fibr6es
de
conjecturer
En
plus
si
T > 0 alors
proposition
le
~ @.
m
[wr(F)]
donne
(r-0+g).
(3)
dans
le
d'existence" (a)
naturel de
- e(F).sg-~.0g-T cas de
du probl~me de p o s e r
rang
plus
classique
de
I, d = qui
Brill-Noether.
la q u e s t i o n
grand
(s =
Kleiman-Laksov
que
un.
(b) m ~ m e
0)
corre-
Ii n o u s dans
Pr~cisement
le
on p e u t
que:
Conjecture Soit calement
X un c o u r b e libre
sur
dim
~ modules
X de
Wr(F) m
d~gre =
g~n~raux d,
de
rang
et
Fun
faisceau
set
g~n~ral.
g- (r+1) ( r - s m - d + ( s - 1 )
(g-1)+g).
io
Alors
209
REFERENCES [1] G h i o n e
La conjecture
F.:
ces r~gl~e8. try, 40, [2] G h i o n e
63-79
of the W e e k
1980,
Zur Math.
Geomeband
(1981).
di Napoli) P.,
les surf~
of A l g e b r a i c
Teubner-Text
Quot scheme over a smoot curve.
F.:
[3] G r i f f i t h s
Proc.
Bucharest
de B r i l l - N o e t h e r pour
serie
Harris
III, J.:
n.
33,
(Preprint
Univ.
(1982).
On the variety of special
linear
systems on a general algebraic curve. Duke Math. Journal [4] K l e i m a n
S.,
47,
233-272
Laksov
D.:
special divisors.
Franco
Ghione
Dipartimento Universit~
di M a t e m a t i c a
di R o m a
R o m a - Italy
II
(1980).
Another p r o o f of the existence of A c t a Math.
132,
163-176
(1974).
On the Construction
of Rational
Surfaces with Assigned
Singularities
Silvio Greco and Angelo Vistoli
Introduction
In this paper we deal with the following problem, this century:
which was very popular early in
given a singular point x
of the complex ana]ytic surface X, does o there exist a rational algebraic surface Y with a singular point y such that the o germs (X,x o) and (Y,y.) are isomorphic? o TNis problem is related with the attempt of generalizing to surfaces the parametrization techniques which are well known for curves. Much work in this direction was done by several classical Enriques,
C. Segre, Hensel, Jung and others
a historical
such as Del Pezzo,
(see / E/, book 4, ch. 4, section 39 for
account).
Later Franchetta tive answers,
authors,
came back to the problem and provided a number of significant posi-
by using double coverings of the plane
(see / F] /, / F 2 / ) .
In this paper we try a different approach and we get some further positive examples. The general idea, suggested by R. Hartshorne,
is the following:
let X + X
o
be a reso
lution of the normal singular point x , with exceptional carve E. Embed E in a ratio o nal surface Y in such a way that the two embeddings are equivalent (i.e. E has biholomorphic neighbourhoodin larity eqdivalent Two difficulties
either embedding).
in a surface Y • o o arise: how to find E C Y ,
Then E C Y can be collapsed
to x
and when is Yoan algebraic
In section 2 we show that if E is irreducible embeddings of E are equivalent provided
and nonsingular
question in a number of cases,
of genus g, then two
answer to the embedding
listed in 2.5.
In section 3 we give a contractability the previous results,
surface.
that the two normal bundles are isomorphic
and their degree is less than 4 - 4g. This implies a positive
equivalent
to a singu-
criterion,
which allows,
to show that if E is a nonsingular
to a singularity of some rational algebraic
in connection with
elliptic curve,
surface.
then x
o
is
211
I:
1.1
Preliminaries
Let X, Y be two complex analytic surfaces, and let x g X, y O
g Y be two normal o
singular points. We say that the singularities (X, x ), (y, y ) are equivalent if the correspon0
d i n g germs o f a n a l y t i c O--y,y
0
spaces are isomorphic, i.e.
i f ~ , x ° i s i s o m o r p h i c to
as a ~-algebra. o
This notion can he studies by means of the equivalence of embeddings.
1.2
Let E be an analytic curve, and let i:E + X, j:E + Y be closed embeddings in two analytic surfaces X, Y. We say that i and j are n-equivalent if there is an ison n morphlsm of nonreduced analytic spaces (E,O_x/I)~(E,O__y/J ), where I, J are the ideals of the embeddings. We say that i and j are equivalent, and we write i~j, if thay are n-equlvalent for all n >0.
1.3
Theorem: Two normal singularities (X,Xo) , (Y,yo) are equivalent if and only if there are resolutions of x
and y o
such that the exceptional curves are isomoro
phlc (as reduced analytic spaces) and their embeddings are equivalent. See / L
1.4
/ for a proof.
We say that a singularity ( X , x )
belonss to a rational surface (resp. to a ra-
O
tional algebraic surface) if there exists an equivalent singularity ( Y , y ) with O
compact and rational (resp. projective and rational).
II: Equivalence of embeddings
From now on C is an irreducible smooth projective curve of genus g. We give criteria for the equivalence 6f certain embeddlngs of C, with some application to our proble~
If C ÷ X is an-embeddlng in a nonslngular surface, we denote by NX/C the normal bundle, which is an invertible sheaf on C. Recall that the degree of NX/C is C 2, the self-
212
intersection of C in X.
2.1
Proposition:
Let i: C ÷ X, j: C + Y be embeddings of C into smooth analytic
surfaces. Then: (i)
if i~j, then NX/C = Ny/C (as vector bundles over C).
(ii)
the converse is true if deg NX/C < 4-4g.
Proof:
(i) NX/C = (I/I2) v ("v" means "dual"), where I is the ideal of the
embedding. (ii) By / L / , --
th. 6.8 (with A. = A.
--
1
= C and s
i
= 2), it is sufficient to O
show that i and j are 2-equivalent.°Now
any 2-thickening C' of C is an extension
O÷L+Oc, ~Oc÷O where L = I/I 2 = N Thus i t
sufficies
v = N
xlc
v
Ylc"
t o show t h a t
all
such extensions
Since C i s smooth the isomorphism c l a s s e s
H I (C, v
of such extensions
are classified
by
eL) (see I-H7, III, ex. 4.10). C
-
-
Now by Serre duality and the degree condition hence there is only one extensions
2.2
are isomorphic.
we have h I (~C ~ mc~L v) = O;
(up to isomorphism).
Proposition:
Let i: C ÷ X be an embedding, where X is a smooth projecfive surP face, and let P e C. Let X' + X be the blow up with center P, and let j: C ÷ X'
be the embedding of C as Proper trasform. Then N x'/c:Nx/c % ~
(-P)"
Proof: Let H be a very ample divisor
on X. We may assume that 0 (C+H) is gene--X rated by global sections (/ H /, 5.17), so that there is an effective divisor D, linearly equiHalent to C+H, and such that P ~supp
(D). Since C is linearly
equivalent to D-H, we have: NX/c ~ O_C (C. (D-H)) Moreover if E = p-l(p) we have j (C) ~ p~(D-H)-E, whence NX'/C = ~c(C'(P
(D-H)-E))
= O_c(C'(pX(D-H)))~ ~ ( - C ' E ) = NX/C~ ~C(-P) which is our claim.
213
2.3
Proposition: Let i: C ÷ X, j: C ÷ Y be two embeddings of C into smooth sur£a~ ces, with Y projective, and assume deg NX/C < 4-4g, deg Ny/C k 3-3g. Then i is equivalent to an embeddifig j': C ÷ Y' where Y' is ohtained from Y by ~lowing up a finite numbe~ of points.
Proof: Put NX/C = ~ ( D ) ,
Ny/C = ~ ( B )
for suitable divisors D and B. We have
h ° (B-D) ~ deg(B-D)-g+l > 3-3g-4+4g-g+l = O. Thus there is a divisor A ~ 0 such that A ~ B-D, i.e. B ~ A+D. Let Y' + Y be the blow up of all the poi6ts of A. Bye2.2 we have Ny,/c = = Ny/c~ ~ ( - A )
2.4
= NX/C, and the conclusion follows by 2.1.
Theorem: Let ( X , x ) be a two-dimensional normal singularity and assume that there o is a resolution X ÷ X with exceptional curve C, and C 2 < 4-4g. Assume further that C can be embedded into a projective rational surface with selfvintersection 3-3g. Then ( X , x ) o
belongs to a rational surface.
Proof:By 2.4 the embedding C ÷ X i s equivalent to an embedding C ÷ Y, with Y rational projective.
Since Ny/C is negative there exists a contraction of C in
Y ~see L-L_Y, 4.9). The conclusion follows from i.3.
Now we list some cases in which 2.4 can be applied.
2.5
Corollary: The conclusion of 2.4 is true if C 2 < 4-4g and, moreover, one of the following conditions holds: (i)
C is a plane curve;
(ii)
C is hypere]liptic;
(iii) g ~ 7.
Proof:
(i) is obvious.
(il) if C is hyperelliptic, then it is birationally equivalent to a plane curve C' of degree g+2 with an ordinary g-uple point. P. By blowing up P we see that C can be embedded in a rational surface #ith self-intersection and we can apply 2.4.
(g+2)2-g 2 > O,
214
(ii) by a thearem of Halphen
(see /-H ~
IV, 6.1) C can be embedded
degree g+3. Let C' be a generic projection
in p3 with
in p2. C' has n = ~(g+2)(g+l)-g
nodes. By blowing up these nodes we can embed C in a rational surface with self-in2 tersection (g+3)2-4n = (g+3)2-2(g+2)(g+l)+4g = -g +4g+5, and it is easy to see that if g ! 7 we can apply 2.4.
2.6
Remarks:
By the above results we can see that:
(i) Two singularities
having a resolution with exceptional
to pl are eq6ivalent
if and only if the self-intersections
any such singularity
is equivalent
ne cone corresponding
bundle.
curves are determined by the normal
Hence any singularity which can be resolved with a non singular elliptic curve belongs
to a rational
section we shall see that it belongs
Ill: Examples of singularities
Proposition:
analytic
surface.
algebraic
surface.
which belong to rational
algebraic
surfaces
criterion:
Let X be a smooth irreducibile
a reduced curve with components
algebraic
E 1 ..... , En. Assume
surface,
= 0
(b) H-D is ample,
for some suitable D = lr.E., r. > O. ii
Then E is algebraically
Proof:
i
contractible.
We use the same technique
as in /-A 7, 2.3.
We may assume that H-D is very ample and that HI(X,Ox(H-D))=
O.
By (a) we have O_o(H ) = ~-D' and hence we get the exact sequence 0
_X (H-D)
-> O
Ox(H)
-> --
->
and let E C X be
that there exists a divisor
H such that H'E
In the next
to a rational
We begin with the following contractability
(a)
it belongs
surface.
of non singular elliptic
curve as exceptional
3.1
are the same. Hence
at the vertex of the aff !
to a rational normal curve in pn. In particular
to a rational algebraic (ii) Embeddings
to the singularity
curve isomorphic
O_D
-+
O
215
which shows that F ( ~ ( H ) )
÷ F(OD) is surjective.
This implies that
base point on D. Moreover H-D is very ample and hence f: X + P
N
IH l h a s a o
IH Idefines a morphism
for some N, such that f(E) is a point, and which is an isomorphism
outside E. Let Y be the normalization
of f(X). Then f factors through X - ~ Y ,
and it is easy to see that this gives the contraction
3.2. Corollary:
of E.
Let C C P 2 be an irreducible
curve of degree c, and let PI' "''' 2 P ~ C with multiplicities e (C) = a.. Let H C~ P be a curve not containing n P. i 1 C and of d e g r e e h , w i t h b i = e p ° ( H ) . A s s u m e t h a t t h e f o l l o w i n g a r e s a t i s f i e d : 1
(i)
~.fi.b. = he
(ii)
b. > a .
i
i
l
(iii)
l
for all
i = 1,...,n
I
F o r /tny i r r e d u c i b l e
c u r v e D C p2 o f d e g r e e
d with
(h-c)d > ~. (b.-a.)r. l
(iv)
h 2- Eb~ + c
i
2
1
e
P. i
(D) = r . we h a v e 1
i
2 - Za. > O.
I
I
Let X be the surface obtained by blowing up PI,...,Pn, transform of C. Then C' is algebraically
contractible
and let C' be the proper in X.
Proof: Let H' be the proper transform H in X, and let us show that H' verifies the assumption of 3.1, with D = C'. By (i) we have H'/~ C' = @; we have to prove that H'-C' is ample. For this we use the Nakai criterion
(see / H /, V, i. I0).
By (iv) we have
( H , _ C , ) 2 = H , 2 + C , 2 = h 2 - Eb.+c 2 2 ~ao2 > O. l l It remains to show that H'-C' intersects every irreducible EI,...,E
be the exceptional
curve D' of X. Let
divisors of the blow up. If D' = E., then (H'-C').
n
l
E. = b.-ao > 0 by (ii). i
i
i
of an irreducible Then (H'-C').D'
If D' # E. for all i's, then D' is the stict transform I 2 curve D of P • Put d = deg D, r i = ep°(D).
= H'.D'-C'.D'
= H.D- ~b.r°-C.D+ lair i =l(h-c)d-~(bi-ai)ri > O i
i
by (iii).
3.3
Theorem:
Let ( X , x ) be a normal two-dimensional singularity. Assume that there o exists a resolution X ÷ X whose exceptional curve C is elliptic and nonsingular.
216
Then (X,Xo) belongs to a rational algebraic surface.
Proof: We embed C in p2 in such a way that we can apply at the same time 3.2. and 2. I. Let s = -C 2 and let NX/C = ~ ( - D ) .
We may assume that D is in the form
QI+...+Qs , where the Qi's are all distinct points: if s = I, then D is just a point, and if s > I, I D I has no base points, and we can apply Bertini's theorem. Now we distinguish three cases, according to the residues of s mod. 3. Case I: s = 3r. By a classical theorem (see / C /, pp. 132-13~ IDI has a divisor of the form 3rP, where P is a point. Embed C in e2 by IBPI. Then IDI = IBr~ consists of all the divisors cut out by the curves of degree r. In particular D = C.C r, for a suitable curve C r. Let C 3 be a cubic curve which cuts C in 9 distinct points PI,...,P9 different 3 from the Q.'s, and let H = 3C +3C . i Let Y be obtained by blowing up QI,...,Qs,PI,...,P9. An easy computation proves that the assumption of 3.2 are satisfied. Thus the proper transform of C in Y is algebraically contractible. By 2.2 we have Ny/c = Np2/c(-~Qi-~Pi) = O-c(~Pi - ~Qi - EPi) = NX/C" Now the conclusion follows by 2.1 and 3.2. Case 2: s = 3r-I As above there is a point P such that 3rP~2QI+...+Q s. Embed C in p2 by 13PI, so that 2QI+Q2+...+Qs is cut out by a curve of degree r. Consider the linear system of all the curves of degree r+3 having a triple point at QI' a double point at Q2 and passing through Q3,...,Qs. Its dimension is at least Jr(r+3)+3, while the dimension of the system of the curves of degree r having a double point at QI and passing through Q2 is ½r(r+3)-4, if r > I. Hence we see that, if r > I, these curves cut out on C, outside 3QI+2Q2+Q3+...+Qs~ a 6 g7; and if r = I, it is easy to see that this linear series has just one base 5 point, aligned with QI and Q2' and the remaining points form a g6" Let then r+3 PI+...+P7 be a divisor in this series, cut out by C , such that all the Pl'S are distinct. Let G 3 be any cubic curve cutting out distinct points PI,...,P7, P8,P9 on C.
217
Since 2QI+Q2+...+Qs
is cut out by a curve of degree r, we have that P8+P9~QI+Q2 .
Hence there exists a curve C r of degree r, not containing C, such that C.C r = r+3 r 3 QI+Q3+...+Qs+P8+P9 Let H = C +C +2C . Then deg H = 2r+9, eQ (H} = 4, 1 Qi(H) = 2 for i = 2 .... ,s, ep (H) = 3, and a straightforward calculation shows that H verifies the conditionslof
3.2. The conclusion follows as in the previous
case.
Case 3: s = 3r-2 The argument is similar to the previous one. Start with 3QI+Q2+...+Qs~3rP,
and
proceed as in case 2, with the curves of degree r+2 having a double point at QI and passing through Q2,...,Qs.
References
/ A / Artin, M.: Some numerical criteria for contractaSility of curves on algebraic surfaces, Am. J. Math.
64, 1962, 485-496
C/
Coolidge J.: A treatise on alsebraic plane curves, Dover books in Adv. Math, New York, 1959
E/
Enriques, F. and Chisini, 0.: Teoria geometrica delle equazioni e delle fun~ioni alsebriche, Zanichelli, Bologna, 1934
FI/ Franchetta, A: Sui punti doppi isolati delle superfici alsebriche, Note I e II, Rend. Acc. dei Lincei, 1946 F2/ Pranchetta, A.: Osserva~ioni Rend. Mat. e Appl., 1946
sui punti d6ppi isolati delle snperfici algebriche,
!-H_~ Hartshorne, R.: Alsebraic Geometry, !i~
Laufer, H.: Normal two-dimensional University Press, Princeton, 1971
Silvio Greco Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy
Springer-Verlag, sinsularities,
New York, 1977
Ann. Math.
Stud., Princeton
Angelo Vistoli Istituto di Geometria Universit~ di Bologna P.zza dl Porta San Donato 5 40127 Bologna Italy
This paper was written with the financial support of M.P.I. Ministry of Education).
(the Italian
POSTULATION
DES COURBES GAUCHES par
Laurent GRUSON et Christian PESKINE
Pour classifier caract@riser
les courbes
les composantes
(courbes de degr@
d
(lisses et connexes) irr@ductibles
et genre
santes et enfin construire
explicitement
C'est le programme
son c@l@bre m@moire
sum les courbes
par exemple,
sa classification:
gauches.
qu'une des deux composantes
en effet si la composante
la courbe g@n@rale
regl@e et rencontre sautes irr@ductibles complexit@
4
du probl@me
classification tion complete
16 H30
des surfaces
composantes
cubiques
de surfaces.
(I)
52)
cubique
Si darts ces deux cas la
du schema de Hilbert
existe une classifica-
(une courbe trac@e sur une surface darts sa composante
qu'une pareille
plus pr@cis6ment,
sur la surface g@n@rale
(de dimension
L'6tude des cinq compo-
o'est parcequ'il
Ii semble improbable
en plus grand degr@;
claire-
([4], app.2) d@montre mieux encore la
quartique n'est pss, daus ces cas, g@n@rale ductible).
@chappe
52) dont
apparait
est trac@e sur une surface
d@crit plus haut.
reste claire~
Rappelons
13 H~8
de
(de dimension
composante
lois une g@n@ratrice.
de
de
I1 nous parait maintenant
sont insuffisantes.
on ne trouve nulle trace de l'autre
pour laquelle
de ces compo-
que Halphen pense r@aliser darts
la courbe g@n@rale n'est pas trac@e sur une quartique ment,
Hd g
une courbe assez g@n@rale
qu'il d@veloppe
([4], app.2),
du schema de Hilbert
g), donner les dimension
chaque composante.
certain que les m@thodes
P3 (I), on veut
de
Ces precautions
chance se renouvelle
nous pensons
qu'il existe des
dont la courbe g@n@rale
d'une famille particuli@re, @rant prises,
irr@-
est trac@e
et non class@e,
elles justifient~
Espace projectif de dimension 3 sur un corps alg@briquement clos de caract@ristique nulle.
219
en pattie,
notre
"D@terminer route
interSt pour le probl@me
le plus petit
courbe
de
Hd g
entier
qui suit:
n, d@pendant
est contenue
de
dans une surface
d
et
g, tel que
de degr@
~ n".
Notons G(d,s)
= sup[g(C),
non contenue Th@or@me
A:
G(d,s)
pour
Cc~3
courbe
darts une surface
I)
Si
s(s-1) < d ,
lisse
de degr@
connexe
de plus
g(C)
tune courbe plane
d
et
~ s].
on a
= J + [d(d+s2-~s)-r(s-r)(s-~)]/2s,
0~r<s;
de degr@
= G(d,s)
de degr@
od
d+r
si et seulement
r
par des surfaces
~ 0 si
C
(s)
et
est li@e
de degr@s
s
et
(d+r)/s. 2)
Si
s2-2s
G(d,s)
3)
+ 3 ~ d ~ s(s-1),
= s 3 - 5s 2 + 9 s - 6 g(C)
male trac@e
sur une surface
s2-2s
et
g(C)
+2
= G(d,s),
= 2, on a
= G(d,s)
d'tme
section
de
(i.e.
C~(E)
= O,
5)
Si
d ~ [(s+2) 2 + 2 ] / 3 ,
de solution 5
du th@or@me
E
on a
nor-
s.
si
C
= s3- 5s2+Ss-5,
est la courbe
des z@ros
est tun fibr@ de corr@lation
nulle
et attirer
au probl@me
rapide
G(d,s)
est incomnu.
des m@thodes
utilis@es
pour
l'attention
du lecteur
dams la 4 eme r@gion.
Signalons
que l'in@galit@
se d@duit
directement
nous ne reviendrons
de Clifford
alors
G(d,s) _~ ~ + ( s - ~ ) d - ( . s3 + 2)
faire ici une analyse
(sur laquelle
est arithm@tiquement
= ~ + (s-3)d
~ d < s 2 - 2s +2,
ces r@sultats
v = d - (s2-2s+3);
C2(~ ) = 4).
[(s+2) 2 + 2 ] / 3
@tablir
o~
od C
de degr@
G(d,s)
E(s-1)
Si
Nous voulons
la courbe
si et seulement
4)
du
+v(v+2s-3)/2,
de plus si
Si
on a
qui d@montre
plus)
sum l'absence
que darts cette r@gion le faisceau
220
inversible
0c(S-1 )
est non sp@cial
elle la meilleure possible?
([7],3.3); cette in@galit@ est
C'est le probl@me de l'existence de courbes
de postulation g@n@rale etudi@ par Hirschowitz Ballico-Ellia
(g~d-3),
(o~
entier
e
n
et
mais non r@solu dans cette g@n@ralit@.
Nous appelons indice de sp@cialit@ d'une courbe e(C)
(courbes rationelles)
C
de
P3' et notons
lorsqu'il n'y a pas de confusion possible), le plus grand
tel que
h1(0c(n)) ~ 0.
Ii est clair qu'on a
ed ~ 2 g - 2.
Posons E(d,s) = sup[e(C), pour
CcP 3
courbe lisse connexe de degr@
non contenue darts une surface de degr@
d
et
< s].
Ce nombre est implicitement @tudi@ par Halphen. Th@or@me B:
I)
Si
s(s-q)
on a
2,3)
Si
s 2 - 2 s + 2 ~ 2 ~ s(s-1), on a
4)
Si
[(s+2) 2 + 2 ] / 3
E(d,s) = s + [ d / s ] - a E(d,s) = 2 s ~ 6 .
~ d < s 2 - 2 s +2, on a
E(d,s) ~ [s-7+/12d-3s 2-6s+ I]/2 On voit que pour les 3 premi@res r@gions la valeur de sugg@r@e par
le
th@or@me A.
E(d,s)
Nous avons ici tune majoration de
est celle E(d,s)
darts la ~eme r@gion dont on esp@re que c'est la meilleure possible (on v@rifie facilement que l'in@galit@ fournie pour cette r@gion est aussi valable et est la meilleure possible pour les 2 eme et 3 eme r@gions). Decrivons maintenant les trois mSthodes utilis@es pour @tablir ces r@sultats. La premi@re m@thode d'Halphen. Soient
P : ~ --> P3
l'@clatement d'un point g@n@ral de
le morphisme de projection. q,(p*J(n))
(o~
J
P3 et
q: P-->P2
Halphen propose d'@tudier les fibr@s
est l'id@al d'une courbe de
P3)
sur
P2' en
221
particulier
de minorer leur deuxieme
classe de Chern,
donc de majorer
le genre de la courbe. La m@thode Si
H
de Castelnuovo.
est tun plan,
et si
est l'id@al dans H du groupe de points e+1 H N C, on v@rifie facilement g(C) ~ ~ hq(JH(n)). Castelnuovo propose I donc d'@tudier la postulation d'une section plane g@n@rale de C. La deuxi@me m@thode Par d@finition
de
JH
d'Halphen. e, il existe une extension
0 --> O p 3 ( - e - ~ )
od
N
est un
cI = e + 4 ,
0p3-module
c2 = d
chef quelles
et
--> N v --> Op3 --> 0 C --> O,
r@flexif
c3 = 2g-2-
conditions
de rang 2 de classes de Chern ed.
II est alors naturel de chef-
sum les classes de Chern d'un
flexif de rang 2 entralne l'existence
Op3-module
r@-
d'une section non nulle de ce
module. Comme nous le verrons, quate pour d@montrer permettent
cette derni@re m@thode
est particuli@rement
le th@or@me B; les deux premi@res
de prouver
A
tions de la majoration
sont illustr@es
qui toutes deux
au mieux par les d@monstra-
du genre des courbes gauches
(s = 2
notations)
fournies par Halphen et Castelnuovo.
Na~oration
du ~enre d'une courbe ~auche suivant Halphen.
Une projection naturelle
g@n@rale n'ayant
q,(p*0p3(1))
N = q,(p*J(1))
que des points doubles,
--> q,(P*0c(q))
et consid@rons
--> H V ( - 2 )
(car
de r a n g 2 e t
est r@flexif
h°(~(d-4-e))
= 0, h°(H(d-3-e))
avec nos
l'application
est surjective.
Notons
la suite exacte
0 --> 0 P 2 ( - 2 ) • 0 P 2 ( - 3 ) M
ad@-
--> q, ( P * ~ c )
~ 0
--> O.
c 1 ( H ) = fl - d ) , et
Comme
Hv = M(d-3)
on en d ~ d u i t
h°(~(d-2-e))
~ 4.
Ceci montre
222
que si avec
M
n'est pas semi-stable,
e = (d-4)/2; dans ce cas
on a
C
est intersection compl@te d'ume
quadrique et d'une surface de degr@ Si
N
est semi-stable,
ce qui entra~ne
on a
que pour
d/2
et on a
g(C) = [(d-2)/2] 2.
c~(H(d-3-e)) ~ 0, doric d - 5 - 2 e
~ 0,
H°(N(e)) = 0; on en d@duit
h°(q~(p~0p3(~))(e)) ~ h°(Oc(e+~)), e ~ (d-5)/2
M(d-3-e) = O P 2 ® 0 P 2 ( - ~ ) ,
ceci implique
e~
soit
g ~ (e+~)(d-e-3)
g ~ [(d-2)/212).
l'@galit@
g = (e+1)(d-3-e)
(et comme
Halphen remarque de plus a lieu seulement si
C
est trac@e sur tune quadrique (l'assertion inverse @rant @vidente). En effet, si cette @galit@ a lieu on a
H~(M(e)) = O, donc par dualit@
Hd(H(d-~-e)) = 0; remarquant alors que le module gradu@
• H°(0c(n)) n>o
est engendr@, sur l'anmeau ~oH°(0P2(n))_ par ses @l@ments--de degr@ e+2, on en d@duit H1(N(d-5)) = 0, soit par dualit@ dire ume application surjective
HI(H(1)) = 0, c'est
H°(q~(p~0p (~))(I)) --> H ° ( 0 C ( 2 ) ) - > O
3 qui montre bien que
C
est trac@e sur une quadrique.
MaCeration du ~enre d'une courbe Gauche suivant Castelnuovo. Pour un groupe de points s@paration de l'id@al de on a sur
raP2,
T, le hombre
r.
Castelnuovo introduit l'indice de
M r = max[n,hl(jr(n)) ~ 0], off Jr
Ii remarque que si
r ne contient pas
est
3 points align@s
XF ~ [d°r/2] - fl et montre dans ce cas, par une r@currence simple d°r, que
r~lhl(jF(n) < (r+1)(d°r-r-3)
pour
r+l < Xr.
Appliquamt
ce r@sultat a ume section plane g@n@rale d'une courbe gauche, on trouve g ~ (e+fl)(d-e-3)
en v@rifiant imm@diatement
est le groupe de points section de
C
e+l ~ XC~H
par le plan
Castelnuovo ne permet pas de montrer que pour
e~ 1
H.
lorsque
CnH
La m@thode de
l'@galit@ n'a lieu
que pour les courbes trac@es sur une quadrique; par contre elle d@montre l'in@galit@ pour le genre arithm@tique d'u~e courbe int@gre. Voyons les difficult@s rencontr@es pour @tendre ces d@monstrations aux cas
s > 2.
des tris@cantes,
Dans la preuve d'Halphen, apr@s l'utilisation du lemme le point crucial est la semi-stabilit@ de
q~(p~J(1))
223
(sauf l'exception bilit@ de
interpret@e).
q,(p*J(n))
(on d@montre alors les intersections pour lesquelles
pour
n<s;
on a bien
et
d'une cubique e = (d-3)/3
n@anmoins montrer que la restriction L
de
e ~ (d-3)/3,
l'insta-
sauf pour d/3
g = I + (e+~)(2d-3e-6)/2),
lorsque
s
grandit.
q,(p*J(n)) n est de la forme ~ OL(ri)
P2
s = 3
et d'ume surface de degr@
mais
devient trop complexe
~ une droite g@n@rale
donc interpreter
si ceci est faisable pour
g ~ (e+1)(2d-3e-6)/2 compl@tes
la classification
Ii faudrait
d'un fibr@
On peut (avec
n<s)
od
[ri, 0 < i < n] est un ensemble connexe d'entiers. Remarquons alors que n E r. = cq(q,(p*J(n))) = ( n + 1 ) n / 2 - d et que c2(q~(P*J(n))) > C~j r.r. i j o l n ([2], l'@galit@ ayant lieu si et seulement si q,(p*J(n)) = ~ OP2(ri)). Cette derni@re
in@galit@
d@montre A.~ pour tout
s
(en l'appliquant
n = s-l) en remarquant
que le genre maximum est donn@ par la suite s-d connexe de r i "d'amplitude maximum" v@rifiant E r. = s ( s - 1 ) / 2 - d . o l Ii est clair que cette in@galit@ n'est la meilleure possible que pour d > s(s-1),
car pour
suite d'entie~s
s(s-1) ~ d
d'amplitude
Darts cette description,
un
ri
gu@re plaisante,
la connexit@
q,(p*J(n))
ne soit pas toujours
de l'ensemble
plutot
on n'a rien dit sum la seule
d'entiers
semi-stable
qu'il s'agira d'une nouvelle variante
l'extension
est de m@me nature
le point crucial
le lecteur devine bien
du th@or@me
qui nous parait moins rebutante.
de la preuve de Castelnuovo
lorsqu'on
est amen@ ~ @tudier des sous groupes
plane g@n@rale.
r
dams
raffinements
de points d'une section
en partie,
le caract@re
est un tel groupe de points,
que
de ce lemme sont surtout p@nibles
Nous voulons tourner,
en introduisant
Rappelons
est le lemme des tri-
d'eventuels
Si
de Grauert-Nulich.
que celle que nous utiliserons
s@cantes;
artifice
[ri] ; bien que
de la m@thode de la section plane, mais nous d@crirons
cette derni@re
un
dans la
maximum.
difficult@,
Cette variante
positif apparait
cette difficult@ par
d'un groupe de points du plan.
contenu dans une courbe
E
de degr@
224
et non darts tune courbe de degr@ moindre, tion de
E
de sommet hors de
l'id@al de d'entier
F
darts
v@rifiant
que ce earact@re,
de
FS c S xP2
S, de groupes
Si
Jr
est
de F la suite ~-~ q.(J~= ~° OPl(-ni). On
de longueur
et du centre de projection. Remarquons g-1 et Z hl(jF(n)) = 1 + E (ni-i)(ni+i-3) I n>l o Soit
une projec-
caract@re
Z
Proposition:
q : Z --> P2
sur une droite ~ l'infini.
Z, nous appellerons
no>nl>...>n~_l>G
v6rifie facilement de
Z
soit
~, est ind@pendant ~-1 qu'on a d°r = °E (ni-i)
une famille int@gre,
de points plans.
Le caract@re
plate au dessus d'une fibre g@n@-
rale est une suite connexe d'entiers. C'est bien s%r celui de la fibre g@n@rique
(le caract@re
m~me si le corps de base n'est pas alg@briquement de celui-ci (off
~
dos); la connexit@
se d@duit du fait que son cSne projetant
est le point g@n@rique
de
S)
Remarque
1:
La section plane g@n@rique
int@gre,
une section plane g6n@rale
courbe est gauche,
est d@flni
est int@gre.
dans
P2(k(~))
([3],3.2).
d'une courbe int@gre
a un caract@re
cette section n'est @videmment
connexe.
@tant Si la
pas align@e,
i.e.
son earact@re est de longueur au moins 2 . L'in@galit@ e+l ~ hq(JHnc(n)) ~ (e+l)(d-3-e), od H N C est la section plane consid@r@e, s'en d@duit imm6diatement. Remarque
2:
Si
C
des trisecantes)
est une courbe gauche lisse connexe,
que la congruence
le groupe des points doubles t@re connexe. minimum a degr@
d-3-e;
le caract@re
par@s par les courbes de degr@ d-4
d'une projection
D'autre part, par d@finition
d-2 = n o > n l > . . . > n d _ 4 _ e _ > d - 3 - e
degr6
D
de ses bis@cantes
(avec
de
de D
e
on salt (lemme
est int@gre;
g@n@rale
donc
a un earac-
une adjointe de degr@
est alors de la forme
n o = d-2
car les points,
s@-
d-3, ne le sont pas par les courbes de
pour une courbe projet@e).
Ii est clair que
d°D>a-~-e(mi-i),_
225
avec
m i = sup(d-2-i,d-3-e),
l'egalit@
on en d@duit directement
ayant lieu si et seulement
il n'est pas difficile
de verifier
si
n. = m. 1
1
que pour
e>l
g < (e+l)(d-e-3),
pour
0
ceci entraSne que la
courbe est trac@e sur une quadrique. Apr@s cette digression,
concluons
se d@duit de la variante
qui suit du th@or@me
ment comme nous avons d@duit, genre des courbes
en signalant
que le th@or@me A 1,2, de Grauert-Mulich
dans la remarque
1, la majoration
exactedu
gauches du fair qu'une section plane g@n@rale n'@tait
pas align@e. Lemme
(Laudal,[5]):
Si
C
est une courbe int@gre de
dans une surface de degr@
L'hypothese nulle de
o~
0
J(t) @Syma~V(-1),
1 'application
t
puis
compos@e
od
~
ment.
prouver
z@ros d ' u n e
nulle,
J
a>>O
tion de
~ expliquer
On d@montre H,
0H(1)) --> J(t) ®OH
la majoration
section
que s i de
a~monc@e s ' e n
t2+ 1 = d
~.(t)
off
~.
comment l a deuxi@me
l e th@or@me de r e s t r i c t i o n Si
E
C.
et pour un plan g@n@ral
H a l p h e n p e r m e t de d @ m o n t r e r l e th@or@me B.
Spindler:
P3"
la
d@duit f a c i l e -
courbe
est
C
est
u n f i b r @ de c o r r @ -
ce q u i d@montre A . 3 .
I1 nous reste
directement
sur
il existe une section non
est l'id@al de
Sym ( 0 H e
fini;
vari@t@ d e s
est
induite.
a un conoyau & support de p l u s
(plans, points)
minimums
0H~ mH(1) - >
On p e u t
t 2 + d > d°C.
est le module des differentielles
exprime alors que pour
alors que pour
lation
t on a
que la vari@t@ d'incidence
Proj(Sym.oV(-1)),
non contenu
t, et si route section plane est con-
tenue clans une courbe de degr@ Rappelons
P3
m@thode p r o p o s @ e p a r
On a p p l i q u e
semi-stable
E est un module r@flexif semi-stable
~ une droite g@n@rale
L
cette
fois
ci
de G r a u e r t - N u l i c h sur
est de la forme
P3' la restric-
226
OL(r i)
I
o~
{ri]
est tun ensemble connexe d'entiers.
Comme corol-
laire imm@diat de cet @nonc@ on v@rifie: (*)
4 c 2 ( E ) - cI(E)2 ~ 0
si
rg(E) = 2, in@galit@ stricte si
E
n'est
pas d@composable, (**)
3c2(E)-c~(E) 2 ~ 0
si
rg(E) = 3.
Si on consid@re l'extension d@crite plus haut 0 --> Op3(-e-# ) --> N v --> Op3 --> 0 C --> O, le th@or@me B I se d@duit imm@diatement de (*). Montrons maintenant que l'in@galit@ enonc@e en B.4 est toujours valable (elle d@montre essentiellement aussi B.2,3).
Ii faut d'abord v@rifier
que cette in@galit@ est @quivalente d l'in@galit@ c1(MV(s-q)) 2 + 2 c 1 ( N v ( s - 1 ) ) + 3 - 3c2(MV(s-1)) ~ O.
Ii reste & d@montrer
le r@sultat suivant: Th@or@me (Hartshorne[8] 0.1): sur et
P3
de classes de Chern
cq ~ -2, on a
Rappelons que positif),
Soit
c i.
un module r@flexif de rang 2 Si
c ~ + 2 c I + 3 - 3c 2 ~ 0
H°(N) ~ O.
X(N) = (cI+4)(c~+2c~+5-3c2)/6+ c3/2
(od
c 3 est toujours
et d@montrons le th@or@me par r@currence sur
d'abord qu'il est @l@mentaire pour l'in@galit@ entra~ne tion
N = 0 2 (-4) P2 l'hypoth@se, donc Comme
N
x(N) ~0,
o2 ~ 0 ,
cI ~ 0 ;
c I.
Remarquons
en effet, darts ces cas
donc l'instabilit@ de
N
ou la d@composi-
(d'apr@s (*)) mais cette d@composition contredit aussi N
est instable et
on peut supposer
d'une extension non scind@e
H°(N) ~ O.
Supposons domc
c I ~0.
h2(N) ~ O, ce qui entra~ne l'existence
0 --~ Op3(-g) --~ E --~ N --~ O, o~
module r@flexif de rang 3~ de classes de Chern
cl
avec
~
est un
I
cq = c I -
i
et
I
c 2 = c 2 - ~ c I.
On v@rifie imm@diatement que l'inegalit@ de l'@nonc@
227 I
implique
c~ 2 - 3c 2 > 0, doric E est instable.
cation non nulle
0p3(n ) --> E
avec
n > (cq-~)/3 ~ -I, elle induit
@videmment une section non nulle de tion non nulle induite nulle
E --> 0p3(n)
avec
Op3(-~ ) --> 0p3(n ) N --> O~3(n)
phisme injectif
N.
Sinon il existe une applica-
n < (ci-4)/3.
o~
on a pris soin de prendre
n
N'
facilement d@montre
n < (o1-4)/3
Sinon, considerons l'homomor-
est le noyau de
minimum,
on v@rifie
Comme l'extension n'est pas scind@e, on a d'autre part
Si l'application
est nulle, il existe une application non
qui d@montre l'enonc@.
N' --> N
S'il existe une appli-
entra[ne
E --> 0p3(n).
c2(N') ~ c 2 + (n+#)(n-cl)-
cq(N' ) = c 1 - # - n
c1(N' ) ~ - 2 .
Comme
< cq;
On prouve enfin
c4(N') 2 + 2 c q ( N ' ) + 3 - 3c2(N') > (n+#)(cl-2n+2) ~ O, ce qui H°(N ') ~ O, par r@currence,
donc
H°(N) ~ O.
Bibliographie [q]
Castelnuovo,
[2]
Elencwajg, G. Ferster, 0.
[3]
G. :
:
Gruson, L. Peskine, C.
- " _
:
[5]
-
:
-
[6]
Halphen~ G.
[7]
Hartshorne,
[8]]
-"
-
Bounding cohomology groups of vector bundles on Pn" Math.Ann. 246, 1980. Genre des courbes de l'espace projectif. Algebraic Geometry. Lecture Notes in Math. n ° 687, Springer.
[LF]
"
Sui multipli di tuna serie lineare ... Rend.circ.Mat. Palermo, t. VII, 1893.
Genre des courbes de l'espace projectif (II). Ann.Scient.Ec.Norm.Sup., 4 e s@rie, t.q5, 1982. Section plane d'une courbe gauche: Postulation. Enumerative Geometry ... Progress in Math. Vol. 24. Birkh~user. N~moire sur la classification des courbes gauches alg@briques. ~ v r e s compl@tes t.lll.
R.
On the classification of algebraic space curves. Vector Bundles ... Progress in Math. Vol. 7. Birkh~user. Stables reflexive sheaves II. Invent. math. 66, 165-190 (1982).
Projective Geometry of Elliptic Curves
Klaus Hulek Department of Mathematics Brown University Providence, RI 02912 USA
Table of Contents
0.
Introduction
I.
The elliptic normal curve
II.
An abstract configuration
Cn _C IPn_I
III. Examples IV.
Normal bundles of elliptic space curves
V.
Open problems
O.
Introduction
In this paper I want to discuss some geometric aspects of elliptic curves. Roughly speaking, the paper is divided into two parts.
In the first three
chapters we shall discuss certain properties of elliptic normal curves.
In
229
chapter IV we shall then apply these results to a certain problem concerning the normal bundle of elliptlc space curves of degree 5. The contents in more detail:
Chapter I is concerned with the explicit
construction of functions embedding a given elliptic curve normal curve of degree
n
in
~n-I
C
as a linearly
These functions will essent#ally be
n-fold products of translates of the Welerstrass
o-functlon.
They will be
chosen in such a way that the symmetries of the embedded curve take on a particularly simple form. curve on
Cn
Cn
Cn ~ Pn-I
In particular, we shall see that the
is invarlant under the Heisenberg group
by translation with n-torslon points.
Hn
and that
Hn
operates
In chapter II we shall then
construct an abstract configuration which is associated to the Heisenberg group in prime dimension
p .
It is a generalization of the well-known con-
figuration associated to the 9 points of inflection of a plane cubic. illustrate this we shall discuss the cases
To
n = 3,4 and 5 in some detail.
Chapter IV is concerned with the normal bundle of elliptic space curves of degree 5.
About two years ago Ellingsrud and Laksov classified the normal
bundles of such curves. one-dlmensional
In their result a central role is played by a certain
family of quintlcs in
]P4 "
We want to apply the results of
chapters I to Ill to get some more information about these hypersurfaces.
Our
main result is that these qulntlcs form a linear family of hypersurfaces whose equations are invarlant under the Heisenberg group
H5 .
Therefore,
closely related to the configuration described in chapter II.
they
are
Due to limited
space, we shall have to restrict ourselves to sketches of proofs rather than give full proofs in chapter IV. Finally, we shall discuss some open problems in chapter V.
230
I.
The elliptic normal curve
Cn ~ ~ n - i
In this chapter we want to collect some material concerning the symmetries of elliptic normal curves known.
Cn S ~ n - i "
Practically all of this was classically
An excellent reference is an article by Bianchi [i ] which was published
in Mathematische Annalen in 1880.
There he mainly treats the case of a plane
elliptic cubic and of an elliptic quintic in
~ 4 ' but he also looks at the
general case of an elliptic normal curve of odd degree•
The even degree case
was treated by A. Hurwitz in [ 6 ].
(I.i)
Let
C
be an elliptic curve with fixed origin
~.
Moreover,
let
F = {nl~01+ n2~ 2 : nl,n 2 E ~ }
be a lattice such that
C = C/F .
P
pq
The n-torsion points of
P~l+q~2 = - n
p,q e Z
group
el -n
c £
n -
and
by
identifying
are then given by
; p,q c ~ .
By abuse of notation we can write G
C
The n-torsion points form a sub-
n
with
(i,0)
and
- ~2 n
with
(0,i)
fix an isomorphism
n
n
n
(n = 3)
0 F
~,~
3
~.j~
"~ --$"
we
231
(1.2)
Next we want
a linearly
normal
to describe
curve of given degree.
w a y that the symmetries First recall
explicitly
of the embedded
a set of functions These
functions
embedding
are chosen
curve take on a particularly
that the Weierstrass
o-functlon
C
as
in such a simple form.
is defined by 2
o(z) : =
~H eEr-{O}
It has the property lattice. formulas
( i - ~ ) (~+~2)~ z e-e
that it has simple
With respect
to translation
zeroes by
eI
exactly at the points and
e2
the following
of the fundamental
hold: e1 n I (z + ~ - ) ~ (z)
(i)
o ( z + e l) -- -e e2 n 2 (z+ 2 )
(2)
Here
nI
the curve
O ( z + e 2) = -e
and C
n2
are the period
itself
For what follows and even degree.
o
defines
o(z)
constants a section
.
of the Weierstrass
So let us first fix an odd integer
(z)
: = O(z
Moreover
we define
the following
e : = -e
Pel+qe2) n
.
constants
n-i n2el 2 n
TileI , ~ : = e
between
n e 3 .
we set
pq
On
o E F(0C(@)).
next we shall have to distinguish
O
~-function.
2n
the case of odd For
p,q e
232
Finally we get functions
xm , m ~ Z
Xm(Z)
Next let
n e 4
: =
as follows
2 mn.z ~mom e ±
be an even integer.
~pq(Z)
Om,O(Z)'...'Om,n_l(Z)
•
Then we write
: = o(z
P~l+q~2 n
i ~2 ~ ( ~ l + - n --)) •
Similarly as above we next define constants
- ½(nl
~
~
: =
l+n2
l )
e
, e
: =
e=
2n
e
which give rise to the functions
Xm(Z)
Using the fundamental
: = ~-'m~2em~iz ~m,O(Z).....O&,n_l(Z ) •
formulas
(3)
Xn+m(Z)
for all integers {Xm;m ~ Z n}
m .
Theorem:
x i E r(0c(nO))
= Xm(Z)
Hence we have in both cases defined a set of
which are a product
of these functions
(I.3)
(i) and (2) it is now easy to check that
is justified
The functions
of suitably adjusted by the following
xi
define
n
(x0(z)
: ...
functions
The choice
theorem.
linearly independent
and the map
z~-->
o-functlons.
n
: Xn_l(Z))
sections
233 embeds
C
a,s a linearly normal curve
Cn ~ Pn-i
of degree
n .
If
2~1 E = e n
then the following formulas hold:
(i)
xi(-z) ~ (-l)nx_i(z)
(ii)
xi(z--~)
~ Xi+l(Z)
~2 (iii) xi(z+~--) ~ glxl(z)
Here
~
.
me,ans that equality holds up to a common nowhere . vanishin,g function
independent of
Remarks:
(i)
i .
It is sufficient to know the fundamental formulas
(i) and (2) to
prove this result. (ii)
At
z = 0
formula (i) holds absolutely,
i.e.
~
can be replaced by
equality.
(1.4)
We next want to rephrase the above result in a slightly different terminology.
To do this we consider the vector space
V=
and denote its standard basis by
¢n
We define elements
{em}mE ~ n
by
The automorphisms
o
and
T
c(e t)
: = et_ 1
T(e t )
: =ete t .
do not commute but one finds
, T E GL(V)
234
[o,T]
Definition:
The subgroup
H
Heisenberg
group of dimension
n
=
c GL(V)
£'id v
generated by
-
n .
o
and
The representation
of
T is called the Hn
defined by the
inclusion is called the Schr~dinger representation of the Heisenberg group.
Remarks:
(1)
For a more general definition of the Heisenberg group and its
Schrodinger representation
see Igusa's book [ 7, p. i0].
locally compact group we have just considered (il)
The centre of the Heisenberg group
Instead of an arbitrary
~n
here.
Hn
equals
~n = {em'idv ; m ~ =}
and the group
Hn
is a central extension
1-->
where H
n
o
is
of order
and n
3
T
are mapped to
In fact if p
3
U n --> H _
(i,0)
n = p
with exponent
--> 2Z x 2~ --> i n n
n
and
(0,i)
I
respectively.
is a prime number then
H
p
is the unique group
p .
To rephrase our result we finally consider the involution
1
:
Cn
-->
e
~---> e m
Then theorem
Cn
-m
(1.3) can be expressed as follows.
The order of
235
(1.5)
Theorem:
invariant origin
(i)
The involution
I
leaves the el.l.iptic normal, curve
Cn C- P n - i
(as a curve) and operates on it as the involution with respect to the
@.
(ii)
Similarly the Heisenberg group
Hn
leaves the curve
Cn ~ P n - i
invariant and operates on it by translation with n-torslon points.
Remark:
We can look at the situation from an even more abstract point of view.
The group
Gn
Zn × Zn
of n-torslon points operates on
operation can be extended to an operation of
on
Gn
C
by translation.
~ n - i = ~ (P(Oc(n@))
This
in
the following way
S
S
I Pi~--> I (Pi + P ) i=l i=l
where
+
denotes the addition on the elliptic curve.
We thus get an irreducible
projective representation
p : G n --> PGL(n,¢)
On the other hand, the Helsenberg gtoup
.
H
is a representation group of n
the group n
× Z
n
Theorem
Gn = •
n
× Z
n
.
i.e. each irreducible projective representation of
can be lifted to a linear representation of
H
n
and vice versa.
(1.3) then tells us that the above projective representation of
lifts to the Schr~dinger r e p r e s e n t a t i o n
of
H
n
Z
n
×
n
236
II.
An abstract configuration
In this section we shall describe an abstract configuration which can be associated to the Heisenberg groups
H
where
p a 3
is a prime number.
P (II.i)
First note that there are exactly
are generated by
(0,i)
and
p+l
(I,A) , ~ ¢ Z
subgroups respectively.
•
c Z x Z . p - p P
They
We shall first
P determine all hyperplanes
H E Pp-I
which are invariant under one of these
subgroups. Clearly
T(H)
if and only if
H
=
H
is one of the following hyperplanes
~k = {x-k = O}
.
Note that
= ok(H0) .
Next we shall determine all hyperplanes
A
We first remark that, because of
o(H)
H
=
such that
R
.
o , the equation of any such
the form
Xo+
p-1 Z Ax m= 1
m
=0 m
.
H
must be of
237
It is easy to check that the invariance under
£ T o
is equivalent to
1
.m ~ m (m-A) for
Xm = A I •e
m = 2,...,p-i .
Hence we c a n set
_ 1(p_l) £-k
Xl=e
for some
k ( Z
p
and the other
I's m
then become
~ (m-p) £ - m k X
m
Then the
p
= E
hyperplanes
p-i e-~(m-p)£-,ink
KR
={ Z
x
m= 0
= O}
; k = O,...,p-i
m
are the hyperplanes invariant under
T£o .
Note that
Hk£ = Tk(Ho£) •
We can sum up our results as follows:
Proposition:
For each of the
p+l
subgroups
c ~P x Zp _
hyperplanes which are invariant under this subgroup.
(II.2)
Next we shall again consider the involution
P
there are exaqtly
p
238
I : ¢P ~ >
e
+--->
¢P
e
m
-m
This involution defines a decomposition of
rP =
E÷OE
cP
into eigenspaces
-
where
E+ = <e0,el + ep_l,...,ep_l+ ep+l> 2 2 E- = <e I - ep_l, .... ep_ I- ep+l> . 2 2 Clearly
dimE + = l(p+l)
Lemma:
dimE- = l(p-l)
E- = H 0
Proof: Clearly
and
(i)
n
H00 n...n H0,p_ 1
We shall first prove that
E- _c H0 .
.
Furthermore recall that
EH0£
p-i 7.. l£x = 0 m= 0 m m
where 1 -~'m (m-p) £ m
is contained in this intersection. is given by
239
Our assertion now follows immediately since
hA = j 1( p - m ) p-m
(-m)A = ei m (m- p )£ = hA . m
(ii) To finish the proof we shall show that H0
and
H0A
are independent.
of the hypersurfaces
To do this we have to examine the matrix
i
i
h0
h0
.
0
i
hI
2 lI
.. •
•
½(p+l)
•
h;-i
•
•
pl
h2
0
~
Using the well known formula for the Vandemonde determinant it will be sufficient to see that
(p+l)
of the
lm
t s
are different.
= ¢k (2k-p) h2k
Therefore we look at
¢2k 2 =
It suffices to see that
2k 2 ~ 2A 2 mod
! if
k,A e {0,. "'' p-i 2 }
are different•
p
But this is clearly so since otherwise
p i 2 (k-A) (k+A)
which is impossible.
This finishes the proof.
240
(11.3)
Our next step is to define for all
k,% ~ ~
the subspaces P
Ek£ : = TkoA(E -) .
Lemma:
n ~k'%' = 0
~%
l!f (k,~) ~ ( k ' , % ' )
It will be enough to show that
Proof.
E00 n E_k,_ % = 0
if
(k,%) # (0,0) .
To see this assume that
x =
p-1 7. X m me
¢ E00 o E_k,_ % .
m=O
Since
x c E00
it follows that
(1)
X
= m
On the other hand, since
x ¢ E_k,_ %
-X --In
it follows that
Tko%(x)- ~ E00 .
This is
equivalent to
2mk (2)
If
A = 0
Xm+Ae
and
Hence assume
k # 0 £ @ 0 .
= -X_m+£
it follows innnedlately from (i) and (2) that By (i) it follows that
one gets
x2£ = 0
which because of (i) implies
time for
m = -3%
we find
x4£ = 0 .
x0 = 0 .
Setting
x_2 % = 0 .
Using
x = 0 .
m = -£
in (2)
(2) again, this
Proceeding in this way one finds
x = 0 .
241
(11.4)
We can now sum up the situation as follows:
planes which we have denoted by
~
constructed
of dimension
~£
p2
subspaces
~£
is contained in exactly
common intersection. contains exactly them.
p
p+l
and
~%
We have found
respectively.
p(p+l)
hyper-
Moreover we have
½(P-l) . Now each of the spaces
of the hyperplanes and is in fact their
On the other hand, each of the hyperplanes ~£
of the subspaces
~
and
~£
and is indeed spanned by any two of
In particular we cay say:
Proposition:
The
p(p+l)
hyperplanes
Hk
and
form a configuration of type
~£
toBether with the
2
spaces
Ek£
(11.5)
So far we have said nothing about the relation of this configuration
to the elliptic normal curve
C P
x
~£
Z p -c Gp
.
m
(0)
=-x
-m
(0)
E00 = E-
contains the origin
~ . Hence
goes through exactly one of the p-torsion points of
Since the hyperplanes
~
sub-
Because of
it follows that the (projective) space each of the spaces
(p~l,p(p+l)p)
p
and
~£
are invarlant under some subgroup
it follows that they each contain exactly
p
of the p-torsion points.
On the other hand the hyperplanes are determined by these points. relation is the following:
~;mE m(ml-£~2)+k~2
We can summarize this as follows.
Cp
~p
The exact
242
Proposition: p
Each of the h~perplanes
of the p-torsion points.
fixed subgroup
Z p =- G p
~
and
The union of all
contains all
p2
Hk% p
intersects
Cp
in exactly
hyperplanes belonging to a
hyperosculating points ,, of
Cp
243
III.
Examples
In this section we want to illustrate the results of the preceding two sections in the case of elliptic normal curves of low degree.
(III.i)
n = 3.
In this case
C3 ~ p2
is a plane cubic curve.
Its equation
must be - at least up to a scalar - Invariant under both the Helsenberg group H3 that
and the involution C3
I .
It then needs only elementary considerations
to see
must be given by a cubic equation of type
3 3 2 x 0 + X l + x 2 + ax0xlx 2 = 0 .
i.e. in Hesse normal form. Next we want to describe the configuration determined by of the 4 subgroups
~3 ~ G3
C
in
I .
Each
3 points of inflection and each of
the triangles contains all
9
have (afflne) dimension
hence coincide with the
points of inflection.
The 9
9
subspaces
It is of type
~£
points of inflection.
Hence we get the classically well known "Wendepunktskonflguration" an elliptic cubic.
and
gives rise to 3 invarlant lines, i.e., to a triangle.
Each of these triangels intersects
1
H3
associated to
(94,123).
In the following picture we want to describe how the invariant lines are related to the 3-torslon points.
244 ~
~o
%44
o
(111.2)
n = 4.
H~
[,I, o l
t2,o)
It is well known that the elliptic quartic curve
cut out by a pencil of quadrlc surfaces. look at the
~
S2V = H0(0p3(2)) •
It splits up into a sum of eigenspaces
5 • Vi i=l
where
=
V1
2
2
is
To determine this pencil one has to
H4-module
S2 V =
C4 ~ ~3
2
< x~+ x2,xl+ x3>
V 2 = < x2- x22,x2 I - x~> V 3 = < xlx3,x0x2> V 4 = < XoXl+ x2x3,xlx2 + x0x3> V 5 = < x0x I- x2x3,xlX 3- x0x3>
245
Now
V1 ~ V3
as
H4-modules whereas no other two of the direct summands are
isomorphic.
Clearly the pencil of quadrics which cuts out
under
It can be neither
C4
H4 .
does not lie in a plane.
1 ~2 xi(~(~l÷~-))
= 0
V1
nor
V4
To exclude
if and only if
or
V4
i ffi 0 .
V5 .
and
V5
C4
must he invariant
It cannot he
One then concludes easily that
2 2 Q0 = x 0 + x 2 + 2a XlX 3
Q1 = x21+ x 2 + 2a x0x 2
where
a =
with
z0
el ~2 = ~-~-
The pencil of quadrics
Q = AQ0+ Q1
contains four singular quadrics which are given by
+i = -K,
-+a .
The vertices of these quadric cones can be easily computed to be
S 1 = (0:i:0:-i) S 2 = (0:i:0:i) S 3 = (i:0:-i:0) S 4 = (i:0:i:0)
since
note that
is the intersection of the quadrics
2 x2(z0) 2Xl(Z0)X3(Z0 )
V1
C4
246
Next note that the involution
I
:
e
~--->
defines a decomposition
e
m
-m
V : E-@E +
where
E
= < e l- e3>
E+ = < e 0 , e 2 , e l +
It follows that
SI
is just the point defined by
lie in the plane determined by
E+ .
projection from
2:1
S1
e3> •
defined a
E
whereas the other vertices
This has the following consequence.
The
map
z : C 4 --> C ~ ]PI
where
C
is a plane conic.
The projection
: F(O
~
induces an isomorphism
(2)) -~ E + .
From this one concludes that the branch points of 2-torsion on
C .
In other words, the vertex
the origin and the points related to
~
SI
~
are the 4 points of
lies on the tangents through
by half-periods.
In this way the 16
tangents to the 4-torsion points can be grouped into 4 sets of 4 tangents which all go through one of the vertices.
(111.3)
n = 5.
In this case we have an elliptic quintic
such curve is cut out by quadric hypersurfaces.
C5 ~ ~ 4 "
Again any
Indeed look at the exact sequence
247
0 - - > IC(2 ) - - > Olp4 (2) - - > 0C(2) - - > 0 .
One checks easily that the map
r(Om4 (2)) --> r(Oc(2))
is surjective and concludes that
h0(7C(2)) = 5 .
Hence there are 5 quadric hypersurfaces through to look at the
C .
To determine these we have
H5-module
S2V = F(0~4 (2)) .
One finds easily that
s2v =
where
V
3
is the 5-dimensional representation of
H5
given by
p : H 5 --> GL(V)
7(0) = o 7(T) = "r 2
It follows that the quadrics containing
C
must be of the form
248
QO = x2 + ax2x3 + bXlX4
Qi = °i (Qo) ;
i = i,...,4 .
One then finds that
2f%
Xo~5--7
2 x4(O)
'Xl(O)
Xl(0)x2 (0)
x2(O)
Similarly one sees that
b-
Hence
C
x2(°) xl~Y =..
1 - ~
•
is contained - and in fact equals - the intersection of the quadrics
Q0 = x~ + ax2x 3 - ~a XlX 4 Q1 = x~ + ax3x 4 - ~a x0x 2 Q2 = x~ + aXoX 4 - ~a XlX3 Q3 = x~ + ax0x I - ~a x2x 4 Q4 = x~ + axlx 2 - ~a x0x 3
Next we want to discuss the confi~ura.tion associated to the Heisenberg group H5
and the involution
I . We get 6 sets each consisting of 5 hypersurfaces.
These form what was classically called the 6 "fundamental pentahedra". spaces
Ek£
define
25 skew lines
~£
whose equations are
x _ k = X l _ k + e2£X4_k = e x 2 _ k + x3_ k = 0 .
The sub-
249
Altogether we get a configuration of type
(256,305).
note the following aspect of this configuration.
It is also worthwhile to
The 6 fundamental pentahedra
determine 6 quintic forms, namely
4
Qo = k_noXk m
4 4 ~(m-p)i-mk Qi = ~ ( Z e xm) ; k=Om=0
One checks easily that the quintlc forms Heisenberg group
H5 .
i = i, .... 5.
Q0,...,Q 5
are invariant under the
Although perhaps a bit more tedious one can also check
that these quintics are linearly independent.
On the other hand, it is straight-
forward group theory (see [ 6 ]) that the quintic forms invariant under an affine 6-dlmensional space
rH(0~4 (5)) ~ r(0~4 (5)) .
H5
form
The intersection of
these quintics can easily be shown to consist of 25 skew lines. Hence we get
Proposition: rH(0P4 (5))
The six fundamental pentahedra determine a basis of the space of invariant quintic forms and the 25 skew lines
~£
are the common
intersection of these quintics.
I want to conclude this section with the following remark. C5
from the origin one gets an elliptic normal curve
is compatible with the involution.
It maps
is given by the 1-dlmensional eigenspace of -
i.
Hence
SI
LO0 = P(E-) 1
L00
C5 .
Corresponding to the four quadric cones through L00
plane to
to a point
is the vertex of a quadratic cone through
is the singular line of a rank
This projection SI
which
which belongs to the eigenvalue
that
But
C4 ~ P3 "
If one projects
3 quadric in
C4 .
~4 C4
But this means
which goes through
there are 4 such lines.
is distinguished by the fact that it is contained in the osculating C5
other lines
at ~£
~ . too.
An analogous interpretation can, of course, be given for the
250 IV.
Normal bundles of elliptic space curves
Here I want to discuss a relation between the material presented in sections I to III and a problem concerning Lhe normal bundle of elliptic space curves
C' ~ P 3
of degree 5.
In their paper [4] Ellingsrud and Laksov clas-
sified normal bundles of elliptic quintics
(for a precise statement see (IV.2)).
Their result depends on a certain 1-dimensional family of quintic hypersurfaces YM "
Ellingsrud and Laksov themselves pointed out that it would be desirable
to have a good understanding of these quintics. tion to help towards this goal.
It is the purpose of this sec-
In fact it was by working on this problem that
I was led to study the symmetries of elliptic normal curves. In order to keep this paper to a reasonable length I do not want to give all the details of all the proofs in this section.
I shall, however,
try and outline how to prove the stated results and I trust that this will enable the interested reader to fill in the necessary details himself.
(IV.l)
We start by remarking that every elliptic quintic
projection of some elliptic normal curve
C5 ~ P 4 "
C' ~ P 3
is the
It is the starting point
of Ellingsrud and Laksov's paper to fix some such normal curve
C = C5
and to
classify the normal bundle of a projection according to the centre of projection. If
If
P ( P4-
P ( P4 Tan C
then we shall denote the projection of then the normal bundle of
Np : =
Here
Wp : C ~ >
P3
gPTP3 /T C
Cp
C
from
P
is defined as
.
is the projection map.
We are now ready to formulate the result of Ellingsrud and Laksov.
by
Cp .
251
(IV.2)
Theorem (Ellingsrud/Laksov):
To each line bundle
de~ree 0 one can associate a quintic hypersurface
M ~ Pic0(C)
of
YM ~ I~4 with the following
proper ties :
(i)
Y0 = Sec C
(ii)
YM = YM-I
and
(iii) Each point
(iv)
If
YM # YM'
otherwise
P £ ~4 - Sec C
P ~ YM - Sec C
belongs to a unique hypersurface
then either
(a)
M 2 # 0C
9nd
(b)
M 2 = 0C
then there is an open, non-empty set of points
P ~ YM - Sec C
Np(-2) = M ~ M -I
such that
or if
Np(-2)
to the non-trlvlal extension
Remarks:
(i)
(ll)
YM "
is indecomposable,
EXt~c(M'M)u
isomorphic
@
For a possible definition of the quintics
YM
see (IV.4).
I have been told that Ellingsrud has recently shown the existence
of curves with
Np(-2) = M ~ M
where
M 2 = 0C
but
M # 0C .
As I have said before it is the purpose of this section to gather some more information concerning the quintic hypersurfaces
YM "
We can sum up our
results as follows.
(IV. 3)
Theorem:
i.e. the map PIc0(C)
The qulntic hypersurfaees
M~---> YM
The equations defining the
H 5 , i.e. the (iii)
YM
form a linear family,
consists of the covering induced by the involution on
foll0wed by a linear embedding of
(ii) group
(i)
YM
YM
P1 " are invariant under the Helsenberg
are linear combinations of the fundamental pentahedra.
The intersection of the qulntics •YM
= Tan C u F
YM
consists of two components
252
where
deg Tan C = i0
the 25 skew lines
~
and
F
is another ruled surface of d e~ree 15 cOntaininB
.
We shall devote the following three paragraphs to a discussion of these statements.
(IV. 4)
We shall first recall the definltion of the quintics
YM "
To do this
set
L :
-- OC(H) = OC(SO')
and
V
Moreover,
if
P ¢ ~ 4 ' let
P ~ I~4 - Tan C
Vp c V
:
= H O(L) .
be the corresponding hyperplane.
For any
one has the following commutative and exact diagram over
0
C .
0
Fp --> Vp@0 c --> PcI(L) --> 0
0-->
N*~L-->
t
L
i
J
0
0
Oc~
Here
N
*
= N C/]p4
*
V@0 C --> P~(L) ~ >
0
0C
is the conormal bundle of
C
in
~4
and
~C (L)
is the
253 bundle of first principal parts of
L .
Note that
Fp = Np@L .
In order to vary the point
P
one considers the product
/'a
I'4x C
P4
C .
This gives rise to a diagram
0
0
I
i
F
,
L 0-->
U
l
q* (N* @L) --> q * (V@0c) = p* (V@0~4
4
1
1
0
0
(~)
Applying
q (L@M)
0 ~>
p,
...
4
where the left hand vertical row is exact over this row with
__>
where
F @ q*(L@M)-->
(~4 - Tan C)× C . Tensoring
M E Pic0C , one gets a sequence
q (N*@L2@M)-->
leads to a morphlsm
p 01~4 (i) @ q*(L@M) -->
0 .
254
~M : H 0 (N*@L2@M)
--> 0]p 4
(i) @ H0(L@M)
.
Because of
h 0(N*~gL2~gM) = h 0(I~M) = 5
this map can be viewed as a
5×5
matrix with entries linear forms.
Then
Ellingsrud and Laksov define their quintics as
YM : = {detCM = 0} .
The hypersurfaces
YM
then have the property that
YM = {P; h 0 (Fp~L~M) ~ O} .
We claim that the
YM
form a linear family.
Now if one wanted to use assertion
(lii) of the theorem of Elllngsrud and Laksov one can deduce this fact - at least in characteristic
0 - easily from the fact that a general point lies on
a unique member of this family.
However, we want to choose a different approach
which in fact provides more information. product
P4 × C × Pic0C
and denote by
For this purpose we consider the E
the Poincar~ bundle over
C × pic0C .
Then we have a sequence
0-->
Here
~
F ~ L ~ E-->
N * ~ L2 B E-->
0~4 (I) ~ L ~ E - - >
denotes the tensor product of the pullbacks to
restriction of this sequence to
~ 4 × C × {M}
is Just
0 .
P4 × C × PicOc . (~)
.
Next let
The
255 f : P 4 x C x Pic0C--> P4 x Pic0C
be the projection. free of rank
Then
f,(N * ~ L 2 ~ E)
and
f,(0P4 (i) ~L BE)
are locally
5 and by Grothendieck-Riemann-Roch one finds
A5f,(N*~L28E)
= q*0c(2~)
5 * * A f,(p OP4(l ) ~ L~] E) = p O]P4(1) ~ q*Oc(4~ )
o
From this one concludes readily that the map
: Pic0C--> PN
:
-- P(r(0m4 (5)))
M~--> YM
admits a factorization
PicOc
>
~N linear
~Pic0C/I
= PI
Note that this also gives another proof of part (ii) of Ellingsrud and Laksov's theorem. (IV. 5) of the YM
We next want to outline how to prove the invariance of the equations YM
under the Heisenberg group
are invariant as hypersurfaces.
since
C
is invariant.
H 5 . To do this we first note that the
This follows from their geometric meaning
Hence it will be sufficient to show invariance for
one of the equations, eg. for
Y0 " Remember that
256
YO = {det{o = O}
where the map
#0 : H0(N*~L2) --> 0P4(1) @ H0(L)
was described in the preceding paragraph.
0-->
On the other hand the exact sequence
I~(2) --> Ic(2 ) --> N*(2) --> 0
gives rise to an isomorphism
H0(Ic(2)) ~ H0(N*(2))
In (III. 3) we explicitly found a basis
.
Q0,...,Q 4
of
H0(Ic(2))
.
Using this
basis we find that
det~
= det f~Qih
o is an equation for
Y0 "
Making use of the special form of the quadrlcs
Qi
it is then not difficult to prove invariance.
(IV. 6) YM "
Finally, I want to discuss the common intersection of the hypersurfaces
The fact that the tangent surface
Tan C
is contained in this intersec-
tion was already noticed by Ellingsrud and Laksov [4, §i0] . F
can be described as follows:
The ruled surface
We have seen before that there is a (projec-
tively) 4-dlmensional family of quadrics through 1-dimensional family of quadrics whose rank is
C . Among these there is a 3 .
Then
F
is
nothing but
257
the union of the singular lines of these quadrics, i.e.
F =
Indeed if
P ~ F
on a quadric curve P0
of
Q'
~) sing(Q) Q~C rankQ = 3
is a general point then Q' .
Moreover
where it is smooth.
Cp
.
Cp E ~ 3
is a quintlc curve lying
has a node and passes through the vertex
One finds
Np(-2) = O c ( P o ) e O c ( - P O)
which implies that
hO(Fp@LSM)
for all if
M E Pic0C .
Hence
P E Sec C - (F u TanC)
P
is in the intersection of the
then
Cp
0c ~>
YM "
Conversely,
again is a quintic curve with a node this
time lying on a smooth quadric surface.
0-->
~ 0
Hence there is an exact sequence
Np(-2) --> 0 C ~ >
0
from which one concludes that
h0(Fp®L@M)
if
M @ 0C .
= 0
It should be remarked that the description of
singular lines of quadrlcs is practically contained in [4].
F
as a union of
It was also known
258 to G. Sacchiero. F .
We now want to give a second description of the second syn~netric product
$2C
of the curve
C .
To do this we consider Note that by means of
the map
$2C
-->
C
(P,Q)~---> P+Q
the surface PI-bundle over
C .
over
C
S2C P(E)
becomes a where
PI- bundle over
E
C .
Indeed it is the unique
is an Indecomposable rank
This is equivalent to saying that
$2C
2 bundle of odd degree is the unique
where the minimal self-intersection number of a section
CO
Pl-bUndle is
i.
We shall next construct a map
: S 2 C - - > P4
whose construction is very much inspired by [4] . following convention:
For any point
scalar) unique section in Then we define
Q,R e C
P ¢ C
H0(0c(P)) .
To do this we employ the
we denote by
Now assume that
tp
the (up to a
2PI+ 3P 2 @ H @ 3PI+ 2P 2 .
by
Q ~ H - 2(PI+P 2) R ~ 3(PI+P 2 ) - H
and set
(PI,P2) : = H 0 (L (-PI-P2))tPltP2 • Ct R2 t O3 c V .
259
On the other hand, if
2PI+ 3P 2 ~ H
$(PI,P2)
or
3PI+ 2P 2 ~ H
we set
: = H0(L(-P2))tP2
or
~(PI,P2): = H0(L(-PI))tPI
respectively.
Proposition.
One then has the following
The map
: $2C-->
P4
(PI,P2) ~--> ~(PI,P2)
is a birational morphism of
$2C
onto the ruled surface
F .
It is
I:i
outside the section
D : = ~-I(c) = {(PI,P2); 2 P I + 3 P 2 ~ H}
and the map
~D : D - - >
C
is a 4:1 covering.
Sketch of Proof: that
$
maps
In order to prove this proposition we first have to see $2C
onto
F . To see this let
PI,P2 E C
be two general
280
points.
Then we define sections
in
V = HO(L)
by
2 3 Y0 " = tRtQ
2 2 Yl : = tQtPltP 2 2 Y2 : = tQtRtPltP 2 "
We can then define a quadric
surface
2
Y0Yl-Y2
Clearly
this surface has rank
image under maps
$2C
~
3
of the fibre of
to
F .
= 0 .
and contains $2C
Using techniques
way and hence that
Moreover,
quadric
through
quadrics
To see that D = ~-I(c)
.
through ~
are parametrized
would give a
quadric surfaces
rank 3
quadric
degree 5
curve
in
Finally
~3 "
Cp
Hence
C
2]
arises in this
by
$2C , i.e. that the C
itself.
is birational we want to show that
But this is so because
line of at most one P
C
.
one can see in this way that
there is exactly one such quadric for each fibre of rank 3
(P,P)
line is the
very much like those in [4, prop.
rank 3
is onto.
Its singular
through the point
we can, in fact, show that each
~
C .
each point through
P { C
C .
contained
~
is
i:i
lies on the singular
Otherwise
projection
in the intersection
look at the map
outside
from
of two
261
SD : D m > C
The image of a point
(PI,P2) ~ D
"
is given by
~(PI'P2 ) = P2
On the other hand, for a fixed point
"
P2 E C
the equation
2P 1 ~ H - 3P 2
has four solutions.
This shows that
seeing this goes as follows. four lines of
F
SD
P ~ C .
saying that the quartic elliptic curve
But this is equivalent to
Cp ~ ~ 3
lies on precisely four
The latter we have seen in (111.2).
We finally want to describe how the 25 skew lines in this picture.
Another way of
The assertion says that there are exactly
through each point
different quadric cones.
is a 4:1 map.
To do this let
~
under the canonical projection onto
LD
Lk~
can be found
be the image of the diagonal $2C .
A ~ CX C
Then
= 50
and set-theoretically
~n D = {(P,P); 5P ~ H}
where each of these points has to be counted with multiplicity 2.
Note that
262
if
5P N H
then
¢(P,P)
Proposition:
= P
c
C
.
The imase of the 25 fibres throush
the points
n D = {'(P,P); 5 p ~ H }
under the map
Proof: under
#
~
Let
P E C
~
S ¢ C
the line
Lk£
.
be a five torsion point and let
of the fibre of
we choose a point tion of
are the skew lines
Lp
$2C
through
such that
(P,P)
S # P
.
but
Lp
be the image
In order to describe 2S ~ 2P .
Lp
By the construc-
is given by
Lp = {w 0 = w I = w 2 = 0}
where
5 w 0 = tp 32 w I = tpt s
w2
To see that
Lp
4 = tpt S •
is one of the lines
all those hyper~lanes
Hk
and
Hk~
Lk~
we have to show that
of our configuration
Lp
lies in
which go through
P .
263
But any such hypersurface
is of the form
H = { w = 0}
where
w = tptp+p,'...'tp+4p ,
for some five torsion point span of the sections
w..
P' # ~ .
So we have to see that
w
is in the
To see this note that
l
3 2 2 % w 0 + ~ w I = t p ( l t p + ~ t S) .
Since choose
P # S l
the sections
and
~
t~,t~P P ¢ H0(0c(2P))
such that
Itp2+~t2s = tp+p,tp+4p,
It follows
form a basis.
•
that
w 3 : = tp3t p+p,tp+4p , c span(wo,W I)
Similarly we see that
2 w 4 : ~ tp~stp+p,tp+4p , ~ span(wl,w 2)
Hence we can
264
Finally applying this argument once more to
w3
and
w4
we see that
w E span(w3,w 4) .
This proves the proposition.
V.
Open problems
To conclude this paper I should like to sketch three problems which arise naturally out of the material presented above.
Problem i:
We have just seen that the hypersurfaces
YM
are defined by
quintic forms which are invariant under the Heisenberg group determine a 2-dimensional subspace in
rH(0P4 (5))
The question arises how to describe this subspace. quintics must contain the curve
C .
H 5 ; i.e., they
which has dimension
6 .
One condition is that the
This is not the case for a general
invariant quintic as one can see by looking at the fundamental pentahedra. But it is not
true
to be one of the
Problem 2:
that this is a sufficient condition for an invarlant quintlc
YM'S.
I hope to return to this problem at some later date.
It would be desirable to have a more geometric way to understand
the behaviour of the normal bundle of elliptic quintlc curves in
~3
In
their papers [2,3] Eisenbud and Van de Ven describe how the splitting of the normal bundle of a rational curve can be realized by looking at certain ruled surfaces and cones over plane curves.
It would be very nice to have an analogous
picture for those elliptic quintics whose normal bundle decomposes.
Moreover,
this should lead to an understanding why the normal bundles of certain "special" elliptic qulntics are indecomposable.
265
Problem 3:
This concerns a possible relation between the Heisenberg group in
prime dimension
p
and vector bundles.
If
p = 3
then
H3
is the symmetry
group of a plane cubic and hence is trivially related to the line bundle On the other hand, if
p = 5
then the Heisenberg group
H5
role in the construction of the Horrocks-Mumford bundle Moreover if
C E ~4
F
is the elliptic normal curve and if
surface then there is a section
s c F(F)
This means that one can recover
F
from
plays a central on
~4
Tan C
whose zero-set equals Tan C
(see [5]). is its tangent Tan C .
via the Serre construction.
The question now is whether one can associate to any Heisenberg group some Vector bundle on bundle on
Postscript:
r6
to
~p~-i "
0~2(3) .
H
P
The first step would be to associate a rank
3
H7 .
U n t i l very recently I did not know whether the configuration which
I have described in chapter II and which was first mentioned to me by W. Barth was classically known or not.
At least it seemed to have been forgotten.
It
was only by chance that I found a paper by C. Segre published in 1886 in Mathematische Amalen in which he describes this configuration.
I have now
included C. Segre's paper as number
[8] among the list of references.
Second postscript
In the meantime I have been able to determine
(December 1982):
the affine 2-dimensional subspace family of quintics
YM "
U E rH(~C (5))
which belongs to the linear
It can be written as the intersection of two
3-dimensional spaces, namely
U = r(ITa n C(5)) n FH(I~(5))
.
All quintic hypersurfaces
through the tangent surface
which are
These quintics are closely related to the Horrocks-
H5-invariant.
Mumford bundle
F .
Tan C
have equations
266
References
[1]
Bianchi, L.: Ueber die Normalformen dritter und f~nfter Stufe des elliptischen Integrals erster Gattung. Math. Ann. 17, 234-262 (1880).
[23
Eisenbud, D., Van de Yen, A.: On the normal bundle of smooth rational space curves. Math. Ann. 256, 453-463 (1981).
[3]
Eisenbud, D., Van de Ven, A.: On the variety of smooth rational space curves with given degree and normal bundle, Inv. Math. 67 (1982).
[4]
Ellingsrud, G., Laksov, D.: The normal bundle of elliptic space curves of degree 5 . 18th Scand. Congress of Math. Proc. 1980. Ed. E. Balslev, pp. 258-287, Birkh~user 1981.
[5]
Horrocks, G., Mumford, D.: symmetries. Topology 12,
[6.]
Hurwltz, A.: Ueber endliche Gruppen linearer Substltutionen, welche in der Theorie der elliptischen Transcendenten auftreten. Math. Ann. 27, 183-233 (1886).
[7]
Igusa, J.: 1982.
[8]
Segre, C.: Remarques sur les transformations uniformes des courbes elliptiques en elles-m~mes. Math. Ann. 27, 296-314 (1886).
Theta Functions.
Klaus Hulek Department of Mathematics Brown University Providence, RI 02912 USA
A rank 2 vector bundle on (1973).
~4
Berlin, Heidelberg, New York.
with 15,000
Springer-Verlag
Linkaae
of General
Curves
of Large
Degree
by Robert
Lazarsfeld
and Prabhakar
Rao
Introduction. Our purpose curve
in ~ 3
is to describe
of degree much
we prove a conjecture such a curve
the liaison
larger
than
of Joe Harris
class
of a general
its genus.
([HI, p.80)
can be linked only to curves
In particular,
to the effect
that
of larger
degree
and
are d i r e c t l y
linked
if X
genus.
Recall is residual
that two curves X,Y C ~ 3 to Y in the complete
they are linked succession a method
linkages.
for producing
also called-has ates.
if the ene can be obtained
of direct
ing from simpler
Ap~ry
Macaulay;
ones. largely
generalized
focused
intersection
and Szpiro
[G] proved
[P-S].
M(X)
of X C ~ 3 , a finite module
curves X , Y C ~ 3 coincides
in higher
up to grading
as it is
relation
dimensions
of Ap4ry
who studied
it gener is linke~ Cohen-
was proved
and Gaeta was
the deficiency
module
~ H l ( m 3 ,~x(n)) ne~
over the homogeneous
Specifically,
are linked
liaison,
start-
if it is Drojectively
The theorem
=
seen as
that a curve X ' C ~ 3
if and only
statement
was
of space curves
on the eouivalence
by the second author,
S = k[T0,TI,T2,T3].
examples
linkage
Later work on linkage-or
the analogous
by Peskine
of two surfaces;
from the other by a
Classically,
interesting
[A] and Gaeta
to a complete
intersection
it was shown
if and only
with either
coordinate in
JR] that two
if the module M(X)
the module M(Y)
ring
of X
of Y or its
268
dual M(Y) v .
Moreover,
deficiency module tially complete the v a r i o u s
any finite S-module M arises as the
of some curve
picture,
in ~ 3 .
Thus one has an essen-
from a c o h o m o l o g i c a l
liaison e q u i v a l e n c e
classes
point of view,
of
that can occur for curves
in ~ 3 .
It is natural understanding
of the curves
In the p r e s e n t smooth
paper,
irreducible
main result
more precisely,
defined
that exist w i t h i n
we c o n s i d e r
curve X C ~ 3
then deg(Y)
of s u f f i c i e n t l y
similarly.
intersection
geometers
apparently
to study space curves
even linkage-i.e.,
linkages-and
liaison
odd linkage, linked to X, then
in an analogous m a n n e r
from the curve
surfaces of lowest p o s s i b l e
here were
one could hope-as did-that
suggested
first raised by J. Harris
of liaison
could be used
by linking a g i v e n curve to a
such an a p p r o a c h
that a general
in its liaison class.
additional
support
Z directly
degree.
curve of lower degree or genus. curves
linked
some of the c l a s s i c a l
techniques
inductively,
that at least for general
as giving
Our
Somewhat
If Y is oddly
(possibly v e r y special)
senses be m i n i m a l
> Pa(X).
curves.
The q u e s t i o n s we c o n s i d e r A priori,
large degree.
of the curve o b t a i n e d by taking the union of
linked to X by i r r e d u c i b l e
(cf [H]).
liaison class.
linked to X, other
We show that if Y is evenly
to X, then it arises
Harris
and pa(Y)
between
an even number of d i r e c t
X and c e r t a i n c o m p l e t e
then,
a given
the linkage class of a general
> deg(X)
we d i s t i n g u i s h
it is a d e f o r m a t i o n
flawed,
for a clearer g e o m e t r i c
(§3) states that if Y is any curve
than X itself,
involving
to ask, however,
curve
Believing
is f u n d a m e n t a l l y
should
in various
Our results m a y be seen,
(if any is needed)
to the
269
philosophy "general"
that there curve.
Harris~s
Some suggestive
conjectures
by Migliore
is no easy way to get one's hands
were obtained
[M]; at least
substantially
results
in the direction
for lines and rational
indirectly
to the present
on a
paper,
of curves
these have contributed as has work of Schwartau
[S].
M o s t of our results X C ~3
subject
surfaces
only
of degree
that hl(X,0X(e))
to the condition
e+4 or less,
~ 0.
§3 to keep the curves seems
likely
are stated
in question results
in the sense of B r i l l - N o e t h e r weak a p p r o x i m a t i o n
to the maximal
We are grateful P. Schwartau also wish
and M.
Stillman
§0.
without
(cf [H], p.79).
J. Migliore, and encouragement.
We
this paper to appear in the
and Conventions.
A curve X C ~ 3
embedded
I x is the ideal (0.2).
general
is even a
though we did not participate
We work over an a l g e b r a i c a l l y
characteristic. one,
It
itself.
Notation
(0.i).
even
for allowing
X C ~3
is missing
rank conjecture
J. Harris,
such
of low degree.
hold for curves What
integer
is used only in
assumption
for suggestions
to thank C. C i l i b e r t o
in these proceedings conference
to L. Ein,
the largest
off surfaces
theory.
curve
that it not lie on any
e being
The generality
that similar
for an a r b i t r a r y
sheaf
points.
field k of arbitrary
is a subscheme Thus X is
of X, and I(X)
If F is a coherent
closed
sheaf
of pure dimension
(locally)
Cohen-Macaulay.
its homogeneous
on ~ 3 , we let
ideal.
270
H,i (m 3
P
F) =
~) n~
H i(~3
i so that H , ( ~ 3, F) is a g r a d e d coordinate 0
ring
S.
We w r i t e
t
~(n))
module simply
r
over
the h o m o g e n e o u s
0 for the s t r u c t u r e
sheaf
~3"
(0.3).
Given
a c u r v e X C ~ 3 , we say that X lies on a s u r f a c e
if F ~ I(X).
If F,G ~ I(X)
curve
scheme
Y whose
meet
structure
properly,
is d e t e r m i n e d
0 ~ P ~ N ~ 0 ~ 0 x ÷ 0 is a l o c a l l y 1 H,(~3,P)
(0.4)
where
0
J NV(-f-g)
ProDn.
Curves Given
minimal a curve
in this
, 0(-f) ~
respectively
in t h e i r X C ]p3 r
section
surfaces
various
senses
minimal
we w i l l
see t h a t X has curve
free r e s o l u t i o n
0(-g) ~
If
of 0 X, w i t h
PV(-f-g)
the d e g r e e s
~ 0
~ 0y
,
of F and G
idea
there
exist vector
liaison
and the n e x t
in its even smaller
is that g i v e n
class.
set
of d e g r e e
to w h i c h
basic
even
= m a x { n l H l(X,0x(n))
not lie on any
any o t h e r
[P-S].
2.5).
e(X)
Our g o a l
as in
link X to a
0y has a r e s o l u t i o n
f and g d e n o t e
([P-S],
§i.
= 0, then
then t h e y
F
is to s h o w that
e(X)
+ 4 or less,
liaison
degree
it is e v e n l y
class.
linked
if X does
then X is in For example,
and a r i t h m e t i c
any two e v e n l y
bundle maps
M 0}.
genus
(Corollary
linked
curves
than
1.5).
X,Y C ~3,
The
0
271 r
r
~9 0(-a i) i=l
which
drop
rank
u
~ E
respectively
if X lies on no s u r f a c e s at l e a s t one allows
and
on X and Y.
of d e g r e e
inequality
us to c o m p a r e
is s t r i c t
geometrically
mild
hypotheses,
odd
linkage
(Proposition
the
integer
e(X)
invariants
of l i a i s o n
is that
from X.
that was
This
of X to t h o s e
statements
We r e m a r k
fact
(Lemma 1.2).
h o w Y is o b t a i n e d
1.6).
, E
+ 4, then b i ~ a i, and
if X ~ Y
analogous
in q u e s t i o n s
v
The c r u c i a l
~ e(X)
the n u m e r i c a l
a n d to d e s c r i b e additional
~9 0(-b i) i=l
of Y,
Under
can be m a d e
the
known
for
importance already
of
to
Gaeta.
We s t a r t
by r e c a l l i n g
curve
as a d e t e r m i n a n t a l
Lemma
i.i.
a useful
representation
of a g i v e n
locus.
Let X C ~ 3
be a curve.
Then there
is an exact
sequence
0 where
, p
N is a v e c t o r
sum of line b u n d l e s
Proof.
u
, N
bundle,
We use a c o n s t r u c t i o n [GLP]
Since
= 0 for n<<0,
R
n
§2).
degree
zero,
cation
of a m i n i m a l
Consider
> -e(X)
similar
free
take
- 4.
to one u s e d by S e r n e s e
the g r a d e d
[Se]
S-algebra
R = H ~ ( ~ 3, 0X).
has
a minimal
generator
to be the
identity.
R necessarily
w h i c h we m a y
, 0,
2 w i t h H , ( ~ 3, N) = 0, and P is a d i r e c t
of d e g r e e s
(cf a l s o
~ IX
S-resolution
of R t h e r e f o r e
The
in
sheafifi-
takes
the
form
0
' P2
~ P1
* P0 ~)
0 •
* 0 x-
~ 0,
272
where each Pi is a direct is the natural map. following
sum of line bundles,
One then obtains,
commutative
and e : 0
~ 0X
using the snake lemma,
diagram of exact sequences
the
of sheaves: 0
0
I 0
0
E
0×
I 1 0
i ~ PI-
P2
~ P0 ~) 0
IT
i N
0
,
1 (~,o)
T
~ Pl
0
1
I
~ 0
~
0x
~ 0
1 0
0
x
1 0
Here N is of course defined H .2 ( ~ 3 I N) follows
as the kernel of Pi;
from the vanishings
By duality one has Ext~(R,S(-4)) summands
of P2 have degrees
of H .1 ( ~ 3
the vanishing I
P0 ) and H ~ ( ~ 3
of '
pl ) "
1 v = H , ( ~ 3, 0x) , and hence all the
h -e(X)
- 4.
Taking P = P2' the lemma
follows. U Remark. M(X)
Keeping the notation
= H ~ ( ~ 3, N).
proof,
observe
Hence the map induced by Pl on global
gives a presentation
,
0 H, ~ 3 ,
p0 )
,
M(X)
,
0
if X lies on no surface of degree e(X) + 3, so that
H 0 ( ~ 3, N(t))
= H0(]P 3, P2(t))
that
sections
of the deficiency module of X:
0 H, (~3, pl ) Moreover,
of the orevious
for t < e(X) + 3, then this is
273
actually
a minimal
a convenient
presentation
method
least determining degree-in
for computing
the number
concrete
suppose
lines L i = {Ai=Bi=0},
complexes
the S-module
a presentation
of generators
this yields
of M ( X ) - o r
and relations
that X is the d i s j o i n t
where A i and B i are linear
is resolved formed
In practice,
at
in each
examples.
For instance,
H ~ ( ~ 3, 0 x)
of M(X).
by taking
the direct
from A i and B i.
of d
forms.
Then
sum of the Koszul
Hence M(X)
given by generators
union
can be described
el,...,e d in degree
as
0, subject
to the relations
d e. = 0 1
i=l A.-e. 1
Using
=
0,
1
the first relation
obtains S(-l) 2d
B..e.
1
a minimal
=
, M~X)
having
, 0.
is treated
ciency m o d u l e
of a smooth rational
form S(-2) 2d-3
surface
lines and rational N o w suppose
an exact
(*)
identically,
union of d complete Similarly,
curve X C ~ 3
J M(X)
discussion
, 0.
presentation
of the
We refer to M i g l i o r e
of the linkage
properties
of
curves. that Y C ~ 3
apply
-,
the defi-
of degree d > 4
is evenly
(0.4) to the exact
linked
to X.
sequence
of
sequence
0
one
the form
has a minimal
, sd-3(-1)
[M] for a more geometric
may r e p e a t e d l y
almost
one of the generators,
The d i s j o i n t
intersections
and not on a quadric
(l
to eliminate
presentation
~ S d-I
0
1
B
, N ~)F
-,
Iy(~6)
,
0
Then we
(i.i)
to obtain
274
for some
6 e ~ , where
On the o t h e r
hand,
B and F are d i r e c t
if A = P ~ ) F,
(**)
0
) A
Our m a i n
technical
summands
of A and B p r o v i d e d
d e g r e e e(X)
uel
~ N~
+ 3 or less.
t h e n of c o u r s e we h a v e
F
lemma allows
sums of line b u n d l e s .
~ Ix
~ 0.
us to c o m p a r e
the d e g r e e s
t h a t X lies on no s u r f a c e s
Specifically,
in the n e x t
of the
of
s e c t i o n we
shall prove
Lemma
i. 2.
In the n o t a t i o n
just
r ~ 0(-ai), i=l
A =
introduced,
write
with
a I < a 2 <_ ... <_ a r
with
b I _< b 2 _< .., _< br.
and r
B = ~i
0 ( - b i),
If X lies on no s u r f a c e s
b. 1
If m o r e o v e r Y ~ X,
> a. --
of d e g r e e
for all
--
X l i e s on no s u r f a c e s > a.
1
l
+ 3, t h e n
1 < i < r.
1
t h e n b.
e(X)
--
of d e g r e e
for at l e a s t one
index
e(X)
+ 4, and
if
i.
r [ (bi-ai). Observe i=l a l s o t h a t at l e a s t w h e n F = 0 so t h a t A = P, the len~na is h i g h l y N o t e t h a t the i n t e g e r
plausible: submodule generators
for t h e n
(*) is j u s t the
the h y p o t h e s i s
H,0(]P3, P) C H,0(~ 3, N)
sum
on X i m p l i e s
consists
t h a t the
free
of the l o w e s t d e g r e e
of H 0 ( ~ 3, N).
Before proceeding,
Corollary
6 in
1.3.
we n o t e the
Let X C ~ 3
amusing
be a c u r v e n o t l y i n g
on any s u r f a c e of
275
degree
e(X)
+ 4 or
less.
module
is i s o m o r p h i c
to M(X)
n > 0,M(X) (n) c a n n o t
In p a r t i c u l a r ,
Proof.
If Y c ~ 3
X,
the
Lemma
X is d e t e r m i n e d
then Y is e v e n l y
integer
whose
grading),
as the d e f i c i e n c y by
linked
to X
deficiency
and for
module
of any
its m o d u l e .
is a c u r v e w h o s e m o d u l e
M(Y)
6 being
(with the g i v e n
be r e a l i z e d
curve.
to g r a d i n g ,
Then X is the o n l y c u r v e
coincides (by
JR]),
w i t h M(X)
up
and we have
= H,l(m 3, N) (-6) = M(X) (-6),
introduced
1.2 a s s e r t s
that
above.
But u n d e r
6 > 0 unless
the h y p o t h e s i s
on
Y = X.
Remarks. (i)
By c o n t r a s t ,
below,
using
Schwartau
[S] has
any n < 0, t h e r e M(Y) (2)
We shall
In this
check
that g i v e n
infinitely
in
§3 that
w h e n X is a g e n e r a l
form,
the r e s u l t
([H], p.80). by M i g l i o r e
ment
exist
shown
w h i c h we
many
shall r e v i e w
any c u r v e
curves
X C ~ 3 , and
Y C ~3
with
= M(X) (n).
satisfied
(3)
a construction
The
[M] w h e n
At least showing
curve
had b e e n
statement
X is a u n i o n
in a s p e c i a l
case,
t h a t the h y p o t h e s i s
of the result. on a s m o o t h
last
the h y p o t h e s i s
Specifically,
surface
has an e x a c t
S C ~3
of the c o r o l l a r y
of s u f f i c i e n t l y
conjectured
large degree.
by J. H a r r i s
of the c o r o l l a r y
was
there
is a s i m p l e
on X is n e c e s s a r y
suppose
of d e g r e e
geometric
~ 0s(X)
argu-
for the v a l i d i t y
t h a t X is r e d u c e d ,
and
f + 4 < e(X)
T h e n one
+ 4.
-X , 0S
established
of lines.
sequence
0
is
, ~x(-f.H)
~ 0,
lies
276
where f
H denotes
e(X),
duality
vanishing linear
the h y p e r p l a n e shows
are evenly (with the
linked,
class
H0(X,~x(-f.H))
of H I ( S , 0 s ) , it f o l l o w s
s y s t e m on S.
on S. ~ 0.
that X moves
But any two c u r v e s
Since In v i e w of the
in a n o n - t r i v i a l
in s u c h a l i n e a r
and t h e i r d e f i c i e n c y
modules
are
system
isomorphic
same grading).
Our n e x t o b j e c t of L e m m a
that
divisor
1.2.
To t h i s
"basic double
Given
is to i n t e r p r e t end,
we n e e d
geometrically
first
the c o n c l u s i o n
to d e s c r i b e
certain
linkages."
a curve
X C
~3,
containing
X, a n d c h o o s e
any
surface
H of d e g r e e
properly.
If G is a g e n e r a l
surface
t h r o u g h X of s u f f i c i e n t l y
large degree,
let F be a s u r f a c e of d e g r e e
s u r f a c e G, double
and we w i l l
linkage
struction
say t h a t
using
F and H.
of l i a i s o n
addition
theoretically,
Evidently
([S], D.91)
t h a t Da(Y)
that this difference u n l e s s X is a line,
1.4.
- Pa(X)
c a s e of the c o n -
by S c h w a r t a u
+ f-h,
= hf.(h+f-4)/2
is a l w a y s
meaning
Let X C
Then
= deg(X)
non-negative,
on the
f r o m X by a b a s i c
is a s p e c i a l
introduced
deg(Y)
s u r f a c e of d e g r e e ~ e(X) to X.
This
, and t h e n
Y d o e s not d e p e n d
it is o b t a i n e d
F
and
[S].
intersection
it f o l l o w s
+ h.deg(X). and
Set
from
Observe
is s t r i c t l y p o s i t i v e
a n d h = f = I.
The geometric
linked
Y.
Y is the u n i o n of X and the c o m p l e t e
of F a n d H.
Proposition
h > 1 meeting
w e m a y use F and G to l i n k X to a c u r v e X
u s e F and G - H to l i n k X* to a c u r v e
f
~3
of L e m m a
be a c u r v e not c o n t a i n e d
+ 3, and
there exists
X = XI,
X2,
1.2 is s u m m a r i z e d
...
let Y C
~3
a sequence
, Xm_],
xm
in
in any
be a n y c u r v e
of c u r v e s
evenly
277
such
that Xi+ 1 is o b t a i n e d
suitable
surfaces
of X m t h r o u g h
F i ~ I(X i) and Hi,
curves
having
X lies on no s u r f a c e one n o n - t r i v i a l
The
statement
that, linked Proof.
double
any curve
(by v i r t u e
X and Y a r i s e
of
~
exact r ?i i=
must
that
linkage
Moreover
if Y ~ X, then at least
occur.
in F i g u r e
from X as i n d i c a t e d
on X,
the a s s e r t i o n
of Lemma
0(-ai)
u v
~ E
Observe
is e v e n l y
1.2
-~ 0
~ IX
-~ E-------+ Iy(6)
of rank
r+l, w i t h
2(m3,E) H,
, 0,
= 0, and
G
#
i.
sequences
bundle
if
JR]).
r
Basic double
using
Y is a d e f o r m a t i o n
module.
is i l l u s t r a t e d
Y obtained
~=I 0 (-bi) w h e r e E is a v e c t o r
such
+ 4, and
linkage
the h y p o t h e s i s via
and
double
deficiency
e(X)
of the P r o p o s i t i o n
Under
0
a fixed
of d e g r e e
basic
conversely, to X
from X i by a b a s i c
/
y Figure
1
is that
278
~l" = b i - a.z -> 0 for all
We a r g u e implies
by d e s c e n d i n g the a s s e r t i o n
Suppose Given
first
on
6 = [6 i that this
set-up
of the P r o p o s i t i o n .
that
6 = 0, so that
a i = b i for 1 ~ i ~ r.
t ~ k, let
w t = tu + Then
induction
1 _< i _< r.
for g e n e r a l
a curve
X t, and
to form
a flat
Zariski
open
(l-t)v @ H o m ( ~ 0 ( - a
t ~ k, the v e c t o r it is e l e m e n t a r y
family
i) ,E).
bundle
that
of s u b s c h e m e s
set U C A 1 c o n t a i n i n g
of
map w t drops
the c u r v e s ~3,
0 and
rank
X t fit t o g e t h e r
parametrized
i.
along
by a
For t @ U one has
1 i?3 M ( X t) = H, ( , E) , and so Y is a d e f o r m a t i o n
of X t h r o u g h
curves
with
fixed d e f i c i e n c y
module.
Assuming
then
that
of u and v r e s p e c t i v e l y ,
6 > 0, let u i and v i be the and d e n o t e
by s i the
i th c o m p o n e n t s
i m a g e of v i in
H o m ( 0 ( - b i),I X ) :
o (-b[) I N r 0
As b e f o r e in i.
we
u= (u I, . . . ,u r)
, ~ 0(-a i) i=l
suppose
that
the
~v i
~ i
~ E
integers
{a i} and
* Ix
{b.}
+ 0.
are n o n - d e c r e a s i n a
279
Let
£ @
Re-indexing
[l,r]
the
or b£+ 1 > b~.
be the
largest
{b i} if necessary, We a s s e r t
that
for
such
that
6£ > 0.
we m a y a s s u m e
that
either
some
sj ~ H 0 ( ~ 3, 7x(b j ) is non-zero. for
integer
index
In fact,
j ~ £, the
section
one has a i = b i > b£
i > £, and
if s. = 0 for all j < £, then 3 factor t h r o u g h those of u:
of v would
£ = r
the
first
£ components
£ 0 (-b i ) s
s
" i=l
l
£ ~) 0 (-a i) i=l
E (u I ..... u£)
But
[ a. < [ b since i=l z i=l i v, would d r o p rank along rank
exactly
(v I ..... v£)
~, > 0, and
so the m a d
a surface,
whereas
on the curve
If Q is a g e n e r a l
Y.
Hence
form
~, and hence
in r e a l i t y
sj ~ 0 for
of d e g r e e
some
b£ - bj,
also
v drops
j ~ Z, as claimed.
then
the
section
F = s£ + Qsj ~ H0(]P 3, Ix(b£))
is non-zero. double by
Let X 2 be the
linkage
(0.4),
using
X 2 and Y
F and
curve
obtained
a general
(trivially)
from X = X 1 by a basic
surface
are r e a l i z e d
H of d e g r e e
6£.
Then
via
r
0
.; ~ i=l
0(-ai) ~
0(-a£-~£)
, E ~
0(-a£)
, IX2(6£)----~
r
0
These
)
i_~1 0(-ai-6 i) ~
sequences
the proof,
hence
satisfy
the
0(-a£)
conditions
the e x i s t e n c e
', E ~ 0(-a£)
stated
of the d e s i r e d
)
at the b e g i n n i n g sequence
, 0.
Iy(6)
of
of c u r v e s
0
280
follows
by i n d u c t i o n .
follows
from the fact that
e(X)
+ 4, and
Corollary less,
1.5.
at least one
linked
and pa(Y)
to X, then
> pa(X).
We n o w s h o w that u n d e r picture
Proposition of d e g r e e
applies
1.6.
e(X)
Let X C ~ 3
+ 3 or
of the h o m o g e n e o u s Assume
curve
Z.
less.
of d e g r e e
either
e(X)
•
+ 4 or
X = Y or
•
hypothesis,
a
linkage:
Choose
not
a system
lying
on any
of m i n i m a l
surface
generators
(i <_ i <_ £)
ideal of X, w i t h
Z does
of d e g r e e
be a c u r v e
that F 1 and F 2 m e e t Then
of the p r o p o s i t i o r
6 i is n o n - z e r o .
a small a d d i t i o n a l
to odd
F i e I(X)
i.
last a s s e r t i o n
I_ff X lies on no s u r f a c e
> deg(X)
similar
the
if X lies on no s u r f a c e s
if X ~ Y, then
and Y is e v e n l y
deg(Y)
Finally,
c i = d e g ( F i) n o n - d e c r e a s i n g
properly,
so t h a t
not lie on any s u r f a c e
they
of d e g r e e
in
link X to a e(Z)
+ 3 or
less.
R e m a r k s. (i)
If X is r e d u c e d
and
irreducible,
it is a u t o m a t i c
t h a t F 1 and
F 2 meet properly. (2) F 2.
The c u r v e However
Z may depend
on the c h o i c e
if Z' is o b t a i n e d
tion of Z t h r o u g h
curves
with
of the
via F~ and F~, fixed deficiency
surfaces
then
F 1 and
Z' is a d e f o r m a -
module
(thanks
to
(1.4)).
Proof.
Consider
is the v e c t o r
a minimal
bundle
free r e s o l u t i o n
constructed
of H ~ ( ~ 3, N),
from X in Lemma
i.i.
where N
281
Such yields
a resolution
an e x a c t
has
sequence
length
which
2, a n d
we use
sheafifying
to d e f i n e
a bundle
E as
shown:
0
) L2
) L1
.~ L 0
(1.7)
, 0
E
o ~" Thus
, N
each
H ~ ( ~ 3, E)
L i is a s u m of = 0.
""~ o line
On t h e o t h e r
bundles, hand,
and
we have
in p a r t i c u l a r from
(I.i)
an e x a c t
sequence
0
) P
~ N
; Ix
) 0,
where s
P =
Observe
that
(*)
~ i=l
our
0 -d.), 3
hypothesis
d. < c. 3 i It f o l l o w s
0 of H , ( ~ 3, P) m u s t
with
all
d. < e(X) 3 -
on X i m p l i e s
(i < j < s, . . .
in p a r t i c u l a r be a minimal
from
+ 4.
that
1 < i < ~). . (*) t h a t
any minimal
0 generator~ of H ~ ( ~ 3, N).
generator
Therefore
£ L0 =
and
then one
to t h e k e r n e l
(**)
0
sees
~ 0 ( - c i) ~ i=l
that
the bundle
of the natural
, E
)
map
~ 0 ( - c i) i=l
P,
E defined 0 ( - c i)
above ) ~X:
~ 7 X(F 1 .... ,F£)
is i s o m o r p h i c
* 0
282
We n e x t resolution
(***)
of
0
use I Z.
,
to read off
Cancelling
from
redundant
~0(ci-cl-c2) i=3
H ,2( ~ 3, E v)
Since
(0.3)
(**)
a locally
terms,
~ EV(-cl-c2 )
= 0, it follows
free
one
finds:
, IZ
, 0
1 H,(]P 3, 0 z) (=H 2 (]p3 ,
that
Z )) injects
into 3 (~ H, (]p3 r 0 (ci-cl-c 2) ) i=3
Recalling
that
the c i are n o n - d e c r e a s i n g ,
e(Z) On the other
hand
H,0(ID 3 ,
L V 0 (-Cl-C2))
thanks
to
(***),
homogeneous
< c I + c2 -
(c3+4)
surjects onto
ideal
I(Z)
0 H,(]P 3, IZ).
,
Therefore,
less.
Since
,
(1.7)
only
of the
in d e g r e e s
(i ! J ~ s).
so H ~ ( ~ 3, I z) v a n i s h e s
e(Z)
that
and hence,
aenerators
c I + c 2 - dj (*),
from
E v (-ci-c2))
0 = H, (~3, IZ ) can occur
But c I + c 2 - dj ! c 2 2 c I by c I - 1 and
H,0 , ( l m3
onto
that
<_ c I - 4.
H,I(]P 3, N v) = 0, so it follows
c I , c 2 , and
degrees
we d e d u c e
+ 3 ! c I - i, this
in
proves
the P r o p o s i t i o n . Corollary
1.8.
Let X and
1.6,
and
from
Z by a s u c c e s s i o n
tion,
Z be as in the
let Y be any curve
as d e s c r i b e d
Pa (Y) ~ Pa (Z)-
in
oddly
of b a s i c (1.4).
linked
double
statement to X.
linkages,
In p a r t i c u l a r ,
of P r o p o s i t i o n
Then
Y is o b t a i n e d
and then
deg(Y)
a deforma-
> dea(Z),
and
283
~2.
Proof
of L e m m a
We k e e p exact
1.2.
the n o t a t i o n
introduced
in
§i,
so t h a t we h a v e
sequences
0
' P
U
+ N
' IX-
:
0
and 0
* B
v
> N ~)F
H e r e B,F and P are d i r e c t P having
degree
on no s u r f a c e
(2.1)
Note
that this
bundles
implies
(at l e a s t
the s u m m a n d s
we r e p h r a s e
thought
More
~
t h a t the f i r s t
formally,
for a n y
image,
s p l i t as a sum of line
to c o m p a r e
the d e g r e e s
statement
as a d i r e c t
sum of
( s u m m a n d s of H of d e g r e e
so t h a t H = H < £ ~ statement
£ e 7/. t h e r e
H 0 ( • 3, H(-£))
H > £ is its
that
for t < f - i.
the d e s i r e d
splitting
H < £ similarly,
shows
i.e.,
t h a t X lies
of
(1.2). line bundles,
set
H > £ =
and d e f i n e
s u m m a n d of
We a s s u m e
itself
We w i s h
every
F to t h o s e of B.
on ~ 3
integer,
+ 4.
, H0(]P 3, N(t))
that N cannot
of A = P ~
If H is any b u n d l e £ is any
=
f = e(X)
+ 3 or less,
if X ~ ~).
To b e g i n w i t h ,
and
e(X)
H0(]P 3, P(t))
> 0.
sums of line b u n d l e s ,
> - f, w h e r e
of d e g r e e
~ Iy(6)
of L e m m a
H > £,
a n d H < £ = H / H > £.
A moment's
i. 2 is e q u i v a l e n t
is a n a t u r a l m a p
~k0(Z)
~ £),
, H.
to
284
(2.2)
For
ever[
£ e ~,
rk(P
Proof (ii)
of
(2.2).
(i):
in t w o
> £).
cases:
(i)
£ < - f;
and
£ < - f.
We given
have
map
B = B > £~
v ~ Hom(B,N
Hom(B>~,F<~)
=
(*)
0,
suppose
are
vector
line
that
bundles,
morphism
of
only
along (*)
r
i.e
(**)
the
(2.2)
given curve
follows
Let natural
£ ~
rank,
sheaves.
> £~--~
(*).
m _
must
Then
and
N ~F
the
Since > £.
> £
N
is n o t
(thought
rank
v.
But
therefore
B > £ and
since
drop
homomorphism and
< £, a n d
Hence
of
along v
strict
in
as
N ~F
>
a sum
of
a homo-
a surface. fact
drops
inequality
Hence rank
must
hold
<_ r k ( N )
- 1 + rk(F>£).
(**)
has
in t h e
P > £ = P. case
at
But
rk(P)
= rk(N)
hand.
£ > - f. N < £ denote
inclusion
of
same
> £--~ N ~
Y,
B
of
F
+ rk(F>£)
in
- f, o n e
from
0
In v i e w
injection
> £ ~
-,
since
(ii):
B
F = F
injection
holds
bundles)
rk(B>£)
Finally,
the
map
vector the
an
< rk(N)
of
the
so t o o m u s t
get
is a n
equality
bundles
B < £ and
~F)
we
rk(B>£)
Now
Case
proceed
> £) _> r k ( B
£ > - f.
Case
in
We
> £ ~F
(2.1)
P
the
> £~
~ P
and
~ P with
> £
the
cokernel
, N
hypothesis
of the
the given
~ N
composition map
,
< £
£ > - f,
u:P
one
0
has
of
~ N:
the
- i,
so
285
H 0 ( ~ 3 , P>z(t))
when
t < - £ (< f-l).
(*)
Hence
H 0 ( ~ 3, N<£(t))
Consider
e~ , H0(]p3, N(t))
, H0(]P 3, P(t))
= 0
for any t _< - £.
n o w the d i a g r a m
0
B>£ ~ -
0
where
' P>£~)
horizontally
Evidently
we've
Hom(B>~,F<£)
F>£
just
' N ~F
formed
= 0, and
v induces
taking
ranks
an i n j e c t i o n
gives
To c o m p l e t e that
if X lies
if B = P ~
F,
(2.2)
then
Y = X.
decomposition
when
To this
end,
F
since
Hom(P~
F>_f,F<_f)
sum of two exact from
(*) that
B>£e---~ P > £ ~
1.2,
of d e g r e e s
v , N~
observe
F = F>_f ~ F < _ f
~ 0,
seauences. similarly
F>£,
and
£ > - f. W
so that one has an exact , P~
F<£
= 0.
of Lemma
on no s u r f a c e s
u:F<_f and
the
of s h e a v e s
the proof
0
~ N<£ ~
it f o l l o w s
Hom(B>%,N<£) Hence
B<£ ~v
F
it r e m a i n s
f = e(X)
to show
+ 4 or less,
and
sequence , Iy
~ 0,
that v g i v e s
rise via the
to a h o m o m o r p h i s m
, F<_f
,
= 0 for r e a s o n s
of degree,
coker
~ is
286
a quotient locally splits
of c o k e r
free, off
this
from v,
sheaf h o m o m o r p h i s m
v = Iy. implies
Since that
in any e v e n t
coker
n is an i s o m o r p h i s m .
i.e.,
Iy a r i s e s
w:P~
F>_f
as the c o k e r n e l
~ N ~F>_f.
e is Hence
of an i n j e c t i v e
But then c o n s i d e r
the
diagram
I P ~) F > _ f
(*) ~"
0
Since
, P ~
F>_f
H0(]P 3, 7x(t))
an i s o m o r p h i s m
class
We n o w a p p l y of g e n u s
g and d e g r e e
All degrees
C(g)
that
> 2g-i
, 0.
that w f a c t o r s
through
X = Y.
is now c o m p l e t e .
of
curve
large degree. irreducible
To b e g i n with,
Moreover,
when
is i r r e d u c i b l e ,
enjoyed
of
~I to a s m o o t h
d>>g.
< 0.
is n e e d e d
elementary
3.1.
, IX
we a s s u m e
that
d > 2g - 1 the
so it m a k e s
by a g e n e r a l
curve
smooth
sense
curve
to
of g e n u s
d.
of the s u r f a c e s
following
F>_f
(*) shows
whence
the r e s u l t s
s p e a k of the p r o p e r t i e s
Lemma
1.2
of all s u c h c u r v e s
g and d e g r e e
, N ~
of a g e n e r a l
d > 2g - i, so t h a t e(X) family
1
= 0 for t <_ f,
of L e m m a
The l i a i s o n
X C ~3
u Q
as i n d i c a t e d ,
The p r o o f
§3.
W
Fix an such
at this
point
on w h i c h X lies.
estimate
integer
is to b o u n d
f r o m b e l o w the
For our purposes,
the
is s u f f i c i e n t :
g > 0.
Then
that a s u f f i c i e n t l y
there
general
exists curve
a constant
of g e n u s
g and
287
degree
d > C(g)
Proof.
Recall
a general degree
lies on no s u r f a c e s
f i r s t of all
rational
that H i r s c h o w i t z
curve D C ~3
n if and o n l y
of d e g r e e
of d e g r e e
5/~ or less.
[Hi] has
shown that
f lies on a s u r f a c e
of
if
1>
+
i.e., n > _ 3 + /~--J-~.
Now let C C ~ 3 d O ~ 2g-l.
Choose
for w h i c h of ~ 3 and
be any
smooth
of g e n u s
a smooth rational
Hirschowitz's
so that
curve
it m e e t s
theorem
curve
holds.
C at a s i n g l e
g and d e g r e e
D C ~3
of d e g r e e
Translate point
f
D by an a u t o m o r p h i s ~
with distinct
tangents,
let
XO = C U D . Thus X 0 has d e g r e e struction X 0 moves
X 0 lies on no s u r f a c e s in an i r r e d u c i b l e
member
is s m o o t h
degree
d and g e n u s
/6(d-d0)
in s u m m a r y
~enus
(cf
we h a v e
3.2.
of d e g r e e family
Therefore
letting
of
genus
a generic
in ~ 3 smooth
whose
But general
c u r v e of
of d e g r e e
f ~ ~ the
§i a p p l y
g, and by con-
~ / 6 ( d - d 0) - 2 - 3.
of c u r v e s
g lies on no s u r f a c e s
all the r e s u l t s
lemma
follows.
U
in the c a s e at hand,
and
established
Let X C ~ 3
g and d e g r e e
X by two
flat
[T]).
2 - 3, and
Thus
Theorem
d = d 0 + f and a r i t h m e t i c
d>>g,
irreducible
be a g e n e r a l and
surfaces
let of
smooth
irreducible
Z be the c u r v e lowest
degree
directly through
X°
curve
of
linked
to
(Thus
288
for d>C(g),
Z has de~ree
> 4d and arithmetic
genus
> g + 3d(/~-2).)
Then: (a)
X is the only curve with deficiency
given grading),
and for n > 0 there
module M(X)
(with the
is no curve with m o d u l e
M(X)(n). (b)
If Y C ~ 3
through
is evenly
curves with
from X by a sequence
linked
to X, then Y is a deformation,
fixed deficiency of basic
module,
double
of a~curve
linkages.
obtained
In particular,
if
Y ~ X, then
(c)
If Y C ~ 3
curve o b t a i n e d
deg(Y)
> deg(X)
is oddly
linked
and
pa(Y)
> Pa(X).
to X, then Y is a deformation
from Z by a seauence
of basic
double
linkages,
of a and
in particular
deg(Y)
> de 9(Z)
> deg(X)
and Pa(~) Remark.
In case
>_ Pa(Z)
(a) one has the estimates deg(Y) pa(Y)
One can replace to somewhat
> Pa(X),
/~
sharper
> d + 5~
> g +
(7d - 3 / ~ ) / 2 .
in the lemma by ~ bounds
on the degree
(£>0), and genus
and this leads of Y.
289
References [A]
- Ap~ry,
[G]
-
[GLP]
-
[H]
-
R.: Sur les courbes de premiere espece de l'espace trois dimensions, C.R.A.S. Vol 220, 271-272 (1945,1)
Gaeta, F.: Quelques progr~s r~cents dans la classification des v a r i ~ t ~ s alg~briques d~un espace Dr6jectif, Deuxi~ume Collogue de G 4 o m 6 t r i e A l g ~ b r i q u e Liege, C.B.R.M., 1952 Gruson, L., Lazarsfeld, R., Peskine, C.: On a theorem of Castelnuovo, and the equations defining space curves (to appear) Harris, J.: Curves in projective space, L'Universit4 de Montreal, 1982
Les Presses
de
[Hi]
- Hirschowitz, A.: rationelles,
[M]
- Migliore,
[P-S]
- Peskine, I.
[R]
- Rao,
[S]
- Schwartau,
[Se]
- Sernese, E.: L ' u n i r a z i o n a l i t ~ della variet~ dei moduli delle curve di genere dodici, Annali della Scuola N o r m a l e Superiore di Pisa, Vol 8, 3, 405-440, (1981)
[T]
- Tannenbaum, A.: Deformations Vol. 34, 37-42 (1980)
J.:
Sur la postulation g~n~rique des courbes Acta Math. 146 No. 3-4, 209-230 (1981)
Ph.D.
C., Szpiro, Inventiones
thesis,
L.: Liaison des vari6t~s Math. 26, 271-302 (1974)
P.: Liaison among 50, 205-217 (1979) P.:
R. Lazarsfeld Harvard University Cambridge, M a s s a c h u s e t t s
Ph.D.
Brown University
curves
thesis,
in ~ 3 , Brandeis
of space
(1983) alg~briaues
Inventiones University
curves,
Math. (1982)
Arch.
Prabhakar Rao Northeastern University Boston, M a s s a c h u s e t t s
Math.
SOME
PROBLEMS
AND
RESULTS
ON
FINITE
SETS
OF
POINTS
IN
~n
Paolo M a r o s c i a Istituto M a t e m a t i c o Universit& OOIOO
O.
Roma,
Italy
INTRODUCTION
Let closed
PI'''''
field and
ordinate
A
be d i s t i n c t p o i n t s
Ps n
>
ring. Also,
function of s
"G. C a s t e l n u o v o " di Roma
or,
2
let
, and let
A =
S = {b.} 1 i ~ o
equivalently,
in
~n(k)
~ A. l
k
(i > o) denote
, with
the H i l b e r t
, where
is a n a l g e b r a i c a l l y their h o m o g e n e o u s
b. = d i m k A i denote l '
f u n c t i o n or the
co-
the H i l b e r t
"postulation'!
o f the
points. In this p a p e r
we i n v e s t i g a t e
(which is an e x p a n d e d
some r e l e v a n t p r o p e r t i e s
use of a c a r e f u l
study o f t h e i r H i l b e r t
version
o f finite functions,
of a talk g i v e n at the Conference), sets of p o i n t s
in
n
, by making
w h i c h turns o u t to be v e r y helpful
in v a r i o u s cases. M o r e precisely, integers if in
S
t h a t are the H i l b e r t
is a sequence
n
in Section
f u n c t i o n o f a finite
set o f p o i n t s
in
the p r o o f we g i v e p r o v i d e s
function
solution,
S
o n e given in [ M a r ]
[G-M-R
.
Section
for a s e q u e n c e
with Hilbert
function
S
S
n
. Also,
2 c o n t a i n s one basic
>
2
~n
an e x p l i c i t
, in a s o m e w h a t
a generalization
result
(Theorem
as a b o v e w h i c h
of
; moreover,
set o f p o i n t s
n = 2
simpler
2.3)
that gives
force a n y set o f
supported by Consiglio
. We p r e s e n t f o r m than the
of this r e s u l t is p r o v e d
s
Nazionale
in
some s u f f i c i e n t points
to c o n t a i n at l e a s t a fixed n u m b e r of p o i n t s
This w o r k h a s b e e n p a r t i a l l y
sequences
. This p r o b l e m h a s b e e n s t u d i e d r e c e n t l y b y
for a n y
original
conditions
those
in [ R ], where a s o l u t i o n has b e e n given in the case
here the c o m p l e t e
]
(Theorem 1.8)
as above,
having Hilbert
L. R o b e r t s
1 we c h a r a c t e r i z e
in
pn
l y i n g in a
delle Ricerche.
291
subspace o f
~n
or even o n a rational normal curve e m b e d d e d in a subspace of
~n
Also, we give a few a p p l i c a t i o n s of this r e s u l t to the study o f a p r o b l e m o f u n i c i t y for c e r t a i n c o m p l e t e linear series o n subcanonical curves in we s h o w
~n
. In particular,
(Theorem 2.11) that "a n o n - s i n g u l a r irreducible c o m p l e t e i n t e r s e c t i o n o f two
surfaces in
~3
a unique simple
o f r e s p e c t i v e degrees 3 gmn
m,n
w i t h o u t fixed points"
with
m = 3,4
and
n ~ m
, admits
.
The last section is d e v o t e d to the d i s c u s s i o n of two o p e n problems,
closely
r e l a t e d to our study. The first one regards a c h a r a c t e r i z a t i o n o f a complete intersec tion zero-cycle in
~n
, in terms o f the Hilbert function and the so-called
Cayley-Bacharach property
(cf. [ G - H ] 2 ) . The second one relates the "generic"
H i l b e r t function of a finite set o f points in
~n
w i t h the minimal number o f
generators o f the h o m o g e n e o u s ideal o f a set o f p o i n t s in function. of
Finally, in c o n n e c t i o n w i t h a remark m a d e in
1-dimensional r e d u c e d
(Cohen-Macaulay)
standard
~n
h a v i n g that H i l b e r t
[ St ] , we show the existence
G-algebras
(Def. i.i) that
have the H i l b e r t function of a complete i n t e r s e c t i o n and n e v e r t h e l e s s h a v e any p r e s c r i b e d C o h e n - M a c a u l a y type.
T h r o u g h o u t the paper, unless o t h e r w i s e specified, c l o s e d field. Also, t > o , we have
if
{gi } i > o
denotes an a l g e b r a i c a l l y
is a sequence o f integers such that for some
gt = gt+l = ''" = gt+j
for all
j ~ 1
following n o t a t i o n
go
k
gl
"'"
gt
÷
, then we shall use the
292
1.
THE HILBERT
Our main
FUNCTION
objective
Hilbert
function
general
notions
so t h a t R
is g e n e r a t e d
DEFINITION written
uniquely
and
this
Let
k-algebra
1.2
([G-K ] ) :
in the
in
+
~
by
say
R. l
~n
~n
a characterization
(n ~ 2).
First
o f the
we r e c a l l
Let
(i i> o)
be a N o e t h e r i a n
integers
and let
that
R 1 , we
h,i
some
R
is a
say that
be p o s i t i v e
R
R
be a f i e l d
o
G-algebra.
commutative k
,
If m o r e o v e r ,
is a s t a n d a r d
G-algebra
integers.
Then
h
with
m i > mi_ 1 >
..
> mj ~
i-binomial
expansion
of
h
.
can be
form
m
(i x)
expression
R =
Then we
as a
=
IN
is to p r o v e
b y the n o n - n e g a t i v e
k-algebra.
m.
h
section
set of points
([ St ] ) :
graded
is a
SET OF POINTS
results.
1.1
identity R
in t h i s
of a finite and
DEFINITION ring with
OF A FINITE
m.
(i-l) i-i
for
h
+
"'" +
is c a l l e d
(j3)
the
j > 1
. Also,
we d e f i n e
m.+l 1 (i+l)
h =
DEFINITION is c a l l e d
an
co
Now we [Mar
mi_l+l +
(
i
) +
i. 3 ([ St ] ,[ R ] ) :
O-sequence
=
1
state
] or [G-M-R
PROPOSITION
"'" +
m.+l 3 (j+l)
A sequence
O and
of non-negative
=
integers
O
{ci}
, if
and
a simple
Ci+l
property
~<
o f the
ci
for all
function
i >i 1
h + h
(for a p r o o f ,
] ).
1.4
:
i i> o
Let
a,b
be p o s i t i v e
a
<
integers,
b
with
for a n y
a
<
b
i > o
. Then
see
293
COROLLARY
1.5
:
Let
{b.}z i > o
be an
bi+l
The
first basic
for a m o d e r n
proof,
THEOREM let
k
1.6
be a n y {c.} l
(2)
{c.}
.
function ~n
{b.}
. x > o
z >
o
1.7
such
:
that
L ~ B
let
B'
=
~
the H i l b e r t ei + c i - i
(3)
if
e. = ]
(j+n-l) J
for s o m e
(i)
Let
W~ l
L
for s o m e
W., 1
i
linear
as a c o m p o n e n t .
d i m W.z =
(i+n)n - b.l '
we s t a t e
a simple
set of points
for a n y
of degree the
] ;
and
G-algebra,
say
R
=
of system
d i m W~z =
consisting a t t h e m as
the l i n e a r
of degree
i
of
Hilbert
on a hyperplane
. z > o
b i = e i + ci_ 1
then
bi
=
ei +
respectively, B U B'
. Also,
let
system
for
the
linear
, the l i n e a r
k-vector
spaces,
i > 1 <
system
system
in
W. 1
(i-l+n)n
B
by
. Then,
the
linear
for a n y
i ~
j
of
cut out that
w e get:
d i m W['z =
L
j
i <
- ci-i
the c o n c l u s i o n .
cut out on
through
for all
ci-i
o f all h y p e r s u r f a c e s
, whence
n
2) w i t h
lying
{e.} l
(i+n-ln_l) - e~z (el > ei),
W' B,i
denote
'
through
(2)
(n ~
. Then
, then
denote,
pn
~n
lemma.
;
j ~ 1
W? 1
in
function
B U B'
i ~ 1
j > 1
and
Looking
of
d i m W i = d i m W~ + d i m W[' l 1
o f all h y p e r s u r f a c e s
integers
but useful
set of points
Hilbert
we h a v e Let
of non-negative
of a standard
function
cj = c j _ 1
and
[Mac
;
, with
if
W. 1
by Macaulay
are equivalent:
be a f i n i t e
(2)
by
. Then
i > o
was discovered
function
be a finite
>
contain
Also,
result,
b.1
all hypersurfaces L
following
(i)
:
bI = m
= k
o
B and
for all
be a s e q u e n c e
O-sequence
our main
. 19o
with
.
is the H i l b e r t R
m+i (i+l)
O-sequence
i ~ o
the
is an
Let
denote
Proof
on
{ci}
Then
, with
proving
{c.} 1
in
o
.
(i ~ o)
LEMMA
~
o f an
[ St ] o r I T ]
Let
:
z
z
Before
see
field.
(i)
R. l
property
~
O-sequence,
system j
,
294
we g e t
d i m W'B , i =
d i m W~l =
(i+n-l) n-i
(3)
of Lemma
- ei
This
Proposition
(i+n-1) n-1
the n o t a t i o n
introduced
in
(i) a b o v e ,
follows
we have:
directly
t+n-i ( t )
et =
f r o m the p r o o f
for
1 < t <
j
of
(i) a b o v e ,
. This
since by
concludes
the p r o o f
1.7.
THEOREM
1.8
:
Let
S = {b.} i
bo(=l)
T h e n the
following
(i)
The
bl( > 2)
difference
O-sequence
. i > o
be a s e q u e n c e
... b d = s
+
of integers
with
o f the
form
b d # bd_ 1
are equivalent:
b °!
is a n
with
, a n d so we a r e done.
statement
1.4,
; hence,
sequence
{bl}
1 , b{ = b I- bo
=
i ~ o
, ...
of
S
b~l = b i
r
:
- bi-i
-°-
t
; bl-i
(2) ordinate
There
1.9
that
G-algebra Hilbert
s
ring has Hilbert
REMARK implies
exist
:
S
with
in v i e w o f T h e o r e m
Proof r i n g o f the (i) ~ trivial,
:
since
(i)
points
(2)
:
such that their homogeneous
co-
that statement
In fact,
{b~}
.
z
, where
z
X
~
o
following
1.8(1) [St ]
, it is e a s y
above automatically
, if
to c h e c k
is an i n d e t e r m i n a t e ;
hence
R
is a s t a n d a r d
that
S
is the
the c o n c l u s i o n ,
1.6.
(2) ~
s
(k)
observing
function
R [X ]
~ S
O-sequence.
Hilbert of
in
function
It is w o r t h
is a n
function
points
follows
contains
The p r o o f
t h e n we h a v e
from Theorem
a non-zero
divisor
goes by induction b.
1.6,
on
since
the h o m o g e n e o u s
of degree
1
b I i> 2 , the c a s e
= i+l
for
o ~< i ~< d
points
(cf. a l s o [ G - M ] 2 ) "
co-ordinate
, i.e.,
S
bI = 2
being
is the H i l b e r t
1
function So, divide
of
s = d+l
write
the p r o o f
collinear
b I = n+l into
(> 2) a n d a s s u m e
three
steps.
the r e s u l t
true
for
b I ~< n
. We w i l l
295
Step
1
:
Let
the H i l b e r t
function
1
bI
1
(;)
{c.} x
there
exists
...
S1
denote
.n+h-l. ( h )
...
Ch_ 1
CLAIM = c I- Co,
:
...
Proof
The difference , C~l = c i -
(of the Claim)
:
...
(r+n-1) r+l
(8)
, br+l
since
the o t h e r b' r+l the
inequality
~
(r+n) r+l
_
(r+n-l) r+l
<
(n+d-1) d
Ch+ 1
---
Cd_ 1
b' [ r+l
,
we
gives
that
÷
of
that,
S1
~
r+n (r+l)
_
r+n-1) ( r+l
get:
if
, where
C'o = i, c I' =
b'r+l
precisely
the the
(r+n-l)
>
r+l
1.5.
Then,
then
form:
, then
:
r+n (r+2)
+
equality,
is o f
1 < r < h
]
from Corollary
b' r+l
"'"
O-sequence.
If we h a v e of
, such
÷
ch > Ch+ 1
{C!}l i ~ o
b'
=
r ~ h)
[ ... ]
"'"
bd = s
(~)
b' = r+l
r-binomial
for
1 < r < h
from
(~)
we
is c l e a r . (r+n-l) r+l expansion
+ [
"'"
of
(8)
follows
for a n y
immediately. r > o
In fact,
; hence,
from
by our hypotheses, (8)
we g e t
get
If not,
'
the Claim
b r+l '
~
follows
expansion
n+h+l ( h+2 )
Ch
we o b s e r v e
... ~ c h
whence
"'"
r+l
(~)
(I <
the p a r e n t h e s i s
Now, b'r+2
,
in
(r+l)-binomial
hence b' r+l
co ~ c I ~
...
and
is a n
First
(~)
In fact,
Cl
sequence
Ci-l'
bh+ 2
o < h < d-1
... < c h
Co
subtracting
:
(n+h h+l )
, with
by
from
ch
h
the s e q u e n c e
~n
S
obtained
bh+ 1
"'"
an integer
sequence
in
bh
1 = cO < cI <
Let
be the
of a hyperplane
co
Then
. x > o
we have
]
296 b' r+2
and
~
' [br+l
-
r+n-1 ( r+l ) ]
+
r+n (r+2)
,
for
1 <~ r ~< h - i
,
so w e a r e done.
Step
2
:
Define
a new sequence
e
o
T 1 = {e
i
}
by putting
i ~ o
:
= 1
eI = n
eh =
n+h-i ( h )
e h + 1 = b h + I- c h
e h + 2 - b h + 2- c h
e d = b d- c h = s - c h
ed+ j = e d
It is e a s y to c h e c k
t h a t the d i f f e r e n c e
by induction,
exists
with Hilbert
with
there
function
Step
3 :
(I)
cI
Hilbert
hyperplane points
defined
B 1 U B~
n
~n
t h a t g a v e us
S1
= S1
of
points
two cases
is an
T1
~n
in
O-sequence.
Hence,
lying on a hyperplane,
such that
n+l
if we d e n o t e
L n B1 = ~
2 , say
B{
. In t h i s from
c~2)
B1
exists,
by Lemma
, which
case,
by
a set of by
and containing
, then,
S = {b i} i ~ o
starting
Co(2)
:
(such a s e t c e r t a i n l y
as in S t e p
cI
s - ch
. In t h i s case,
is p r e c i s e l y
(II)
sequence
T1
function
of
a set o f
N o w we d i s t i n g u i s h <
j ~> 1
for a n y
proves
we s t a r t
S
(see S t e p
"'"
c!2)3
from
1.7
ch
induction) also
points and by
a set of
, the H i l b e r t
L
~n a
s - ch function
of
the t h e o r e m .
S1
and repeat
the c o n s t r u c t i o n
1 ) . So w e g e t a n e w s e q u e n c e ,
÷
in
(J ~ h
, c (2)j <
say
c h)
S2
.
:
297
Then, not, of
if
c (2) 1
<
we r e p e a t
steps,
n
, we
the construction
we get a sequence,
(t) Co
Now,
by using
completes
the proof
If
S = {b.} z
defined
integer,
c~t)
b~
of dimension
Proof
i.iO
S (I)
where
X
:
way
:
after
1.7 a g a i n ,
If
a finite number
c 1(t) ~< n
with
of Theorem
we a r e done.
This
1.8.
of integers,
:
i
Let
S (d)
S (I) = {i,
b! -
' bi-l'
S
integer
We proceed
let
d >
d-1
with
b
=
{b.}
gl
z ~
,
g2
S
...
d
sequence
is a n y
of
, b~l = h i - b i - l '
S "'"
, } '
a n d so o n . .
.
z
dth-difference
b E = b I- bo, } .
.
d ~ 1
the
= 1 , and
o
o
(d)
"'"
be
a sequence
is a n
O-sequence
gm = O
function of a reduced
of integers o f the
with
b
o
= 1
form
÷
Cohen-Macaulay
G-algebra
standard
by induction
1
and assume function R
f r o m the H i l b e r t
this
the r e s u l t
of a reduced
. Hence
is a n i n d e t e r m i n a t e ,
provides
on
d ~ 1
, the case
d = 1
being
clear,
1.8.
, say
We conclude l.ll(c)
(I) a b o v e ) .
d
is the H i l b e r t
dimension
. Clearly,
÷
and Lemma
denote by
is the H i l b e r t
in v i e w o f T h e o r e m So,
c (t) m
consequence
1
S
.
S2
in c a s e
o
we s h a l l
for a given
Then
:
is a s e q u e n c e
i ~
'
such that
St
hypothesis
a simple
- b~ . . . .
COROLLARY
""
from
given
of the theorem.
in the o b v i o u s
S (2) = {i,
(see t h e a r g u m e n t
starting
say
the i n d u c t i o n
N e x t we s t a t e
positive
are done
function
for
d' < d - i
Cohen-Macaulay
is p r e c i s e l y
. Then,
standard
the Hilbert
by induction,
G-algebra
function
of
of
R I X ],
a n d so we are done.
section with
some useful
S
true
a few general
geometric
of a finite
remarks.
information set of points
In p a r t i c u l a r ,
that can be extracted in
~n
Remark directly
298
R E M A R K l.ll for a n y i n f i n i t e
: (a) field
result by Macaulay
(b) to o b t a i n
It is e a s i l y k
. Also,
(cf. [St,
seen that T h e o r e m 1.8 h o l d s m o r e g e n e r a l l y
Corollary
Corollary
points with
given in the p r o o f o f T h e o r e m
the r e q u i r e d p r o p e r t i e s
steps,
where
of
p o i n t s we get at the end lies on a r e d u c i b l e
s
m
s
is the l e a s t integer
improves an analogous
3.11 ]).
We note t h a t the c o n s t r u c t i o n a set o f
1.10 s l i g h t l y
such t h a t
b
<
m
1.8 in o r d e r
stops e x a c t l y a f t e r
(m+bl-l) m
hypersurface
. Hence
m-i
the s e t
of degree
m
bl-i consisting
of
(c)
m
distinct
Finally,
hyperplanes
w i t h the n o t a t i o n
of
~
(k)
introduced
in the p r o o f of T h e o r e m
1.8,
given
bl-i a n y set of follows
points
in
~
(k)
with Hilbert
from Lemma 1.7 that at m o s t
Moreover, Hilbert
s
this m a x i m u m function
S
is a c h i e v e d
. Similarly,
s - ch
function
o f the
s
in our c o n s t r u c t i o n one c a n d e t e r m i n e
S
=
{bi}
i ~ o
, it
p o i n t s c a n lie o n a hyperplane. of a set of
explicitly
s
points with
the m a x i m u m number bl-i
o f points
t h a t c a n lie o n a subspace
particular,
of any given d i m e n s i o n
the m a x i m u m n u m b e r of c o l l i n e a r p o i n t s
proved by using a straightforward
argument
is
d + 1
(cf. [ G - M - R ])
of
~
(k)
. In
, w h i c h c a n also be
299
2.
HILBERT
FUNCTIONS
In t h i s
section
sets of points
in
series We
on
start
LEMMA
:
Suppose
there
Then
Proof conditions
Let
that
but not at
P
and
take
a form
LEMMA general
2.2
:
Let
position,
following
with
conditions
(i)
b t = tn+l
(2)
b 2 = 2n+l
PI'
"'"
' P
is d u e
s
Pl'
6
r
tn+l
F
Hilbert
least
with
s
such
, for
and
s ~
t ~
j-i
P
H
and
normal
that
bd = s .
no
of
n+l
(h i> i). ]pn
of
the~
, impose
independent (without
independent at
conditions
Pr+l'''''
Ps-(n-h)''''' , which
loss of
proves
P
Ps-(n-h+l)
s
Lemma
2.1.
[H2] ) .
points
. z ~ o
s > tn+2
curve
in
~n
and satisfying
2n+3
lie o n a r a t i o n a l
such
with
, d-i
vanishing
E
r
(see
{b.} z
...
through
be distinct
3
complete
(s > n ~> 2)
]pn-h
(i.e.,
bj_ 1
: for s o m e
finite
we g i v e
b .3- bj -i = n - h
we m a y a s s u m e
impose
. Hence
function
that
r = s-b. 3
any hyperplane
' Ps
1pn
integer
t = i,
. Then,
to C a s t e l n u o v o
"'"
of
Also,
for c e r t a i n
in
position
of degree
FH
properties
functions.
l i e in a s u b s p a c e
' Ps-(n-h+l)
denote P
points
' P
n
result.
points
be the
I 0 ~n(j)l
"'"
H
d
in g e n e r a l
~
"'"
Pr+l'
let
lemma
bt
system
our main
be d i s t i n c t
are
Pr+2'
by our assumptions,
The next
Then
Pr'
stating
s
Hilbert
of unicity
(i ~< j ~< d-l)
o f the
then
given
geometric
IN
~n
let
j
points
Pr+l'
. Now
r
index
s
in
' PS
, and
n-h+2
dependent),
10 ~ n ( J - l ) l
Then,
some
concerning
to a p r o b l e m
before
"'"
OF SETS OF POINTS
certain
curves
{b. } z i~>o
if the
:
results
PI'
to t h e l i n e a r
generality) to
Let
at least
are linearly
(n > 2) h a v i n g
two l e m m a s
exists
In p a r t i c u l a r ,
results
some
subcanonical
2°i
function
we p r o v e
of these
with
Hilbert
PROPERTIES
~n
a few a p p l i c a t i o n s linear
AND GEOMETRIC
.
(s > n >
2)
o n e o f the
in
300
THEOREM
2.3
Hilbert function
Let
:
PI'
be distinct points in
"'" ' Ps
{b.} 1 i ~ o
and let
d
~n
be the least integer
(s > n > 2)
such that
with
bd = s
Suppose that (a)
b d _ I- bd_ 2
(b)
no
Then we have
n-h+l
at least
(2)
if moreover,
: (1) to
Pr' Pr+l'
"'"
(cf.
s
Pr+l'
bd_ 2
q L
r
independent
"'" ' Ps-(n-h+l) we get
some index Pr+l'
j
"'"
(2)
' Ps-(n-h+l)
L
of
to
n
Pr' Pr+l' I 0~n(d-2)l
such that the
impose
' Ps-(n-h+l)'
bd_ 2
, then at least
and moreover
n-h
n
of
, impose independent
to
I0~n(d-2)I
spanned b y
that
. Hence,
p s-(n-h)'''''
Ps
.
"'" ' Ps-(n-h+l)'
Ps-(n-h)
. Hence there exists some index points(1)Pr' to
bd_ 2
it follows
contains
[ (n-h)(d-2)+l ] + (n-h+l)
Pr+l'
"'"
'
. Hence, arguing
Thus, there exists
independent conditions
impose independent
Ps-(n-h)'
10~n(d-2)l
such that the
Ps-(n-h) }
~n
to
points
I O ~ n (d-2)l
,
from Lemma 2.1 that
at least
conditions points
Pr' Ps-(n-h)'
(n-h) (d-2)+l to
10 ~n(d-2)l
(among the
P~ s) l
,
(I) .
This statement
to the one developed
(I)
the points
conditions
contains at least
which proves
conditions
if we iterate the argument above,
points which lie in
pn-h
. Also, we can repeat the argument.
{Pr' Pr+l . . . .
L
n
(without loss of generality)
impose independent conditions
p. 6 L 1
and so on . Then,
Hence
of
r = s- bd_ 1
subspace of
(j # i ; r+l < j < s-(n-h+l))
"'" ' Pj'
the set
with
(see the proof of Lemma 2.1)
r+l ~ i ~ s-(n-h+l),
as above,
pn-h-I
points lie in a
impose independent
(n-h)-dimensional P
s
. Then, we may assume
' Ps-(n-h+l) the
of the
"'" ' Ps
Now, by our assumptions,
Pi'
, and
points lie on a rational normal curve in a
n(d-l)~
(b)), we get
i , with
h ~ 1
bd_n+ I- bd_ n = ... = bd_ I- bd_ 2 = n-h
s
Let
I0
denotes
impose
with
points lie in a
(n-h) (d-l)+2
of the
conditions
L
,
of the
(i)
Proof
n-h
:
(n-h) (d-l)+2
if
=
follows from Lemma 2.2 , by using an argument quite similar
in the proof of
Here the notation
P. 1
(i) above.
means omit
P. 1
301
and
The f o l l o w i n g
result
COROLLARY
:
n > 2
Then,
2.4
Let
, and l e t b2
<
2n
(2)
no
n
o f the
hyperplane,
the
s
2.5
:
(a)
~n-h
is e x a c t l y
, all d i s t i n c t
1
3
6
9
(cf. R e m a r k l.ll(c)).
(c)
Theorem
first case
:
of Theorem
2.3
d+l
of
s
It is e a s i l y as follows
For example,
Yet,
the H i l b e r t
n
h = n-i
, then these p o i n t s
curve
.
in a
actually
lie
points
2.3 are b y no m e a n s n e c e s s a r y
~2
all p a s s i n g
, three p o i n t s o n
in
points
1
7
9
raised
in
a n d five p o i n t s function
function
are c o l l i n e a r 12
the m a x i m u m n u m b e r o f c o p l a n a r
+ points
points
points
for a set
(cf. R e m a r k l.ll(c)) in g e n e r a l ,
2
; hence ii
for
take
the H i l b e r t (cf. R e m a r k
. It is
h ~ n-2 iO
points outside
. Then,
is
2.3(a)
n u m b e r of c o l l i n e a r
number of collinear
and any
points
In this case, w i t h the n o t a t i o n
"minimum"
L
O ,
[ G - M - R ] , w h e r e the
in the f o l l o w i n g manner:
l y i n g in a p l a n e
Ii
L2
the m a x i m u m n u m b e r o f c o l l i n e a r
is no l o n g e r true,
3
12
through a point
f u n c t i o n above d o e s n o t s a t i s f y
to the m a x i m u m
o f the 4
normal
2.2 , s t a t e m e n t
these nine p o i n t s h a v e H i l b e r t
, and our
a w a y that no
this case,
L1
w i t h the g i v e n H i l b e r t
12
lie o n a rational
is also settled.
conic
is
d ~ 4
they contain
on a non-singular
these p o i n t s
that
) or all b u t one lie in a
in v i e w o f L e m m a
three lines in
t h a t such a n e q u a l i t y
3
actually
2.3 gives an a n s w e r to a p r o b l e m
choose
Suppose
s > 2n
~n
n
(a), (b) in T h e o r e m
. Then,
; also,
, we h a v e
worth observing
, with
: "If the m a x i m u m n u m b e r o f p o i n t s l y i n g
b d _ 2 = s-2 = b d _ I- 1
in
pn
function.
of
(of
seen that,
and also
O
, w h i c h is a l s o equal points
in
2.3.
.
from
÷
~n-2
points
and c h o o s e one p o i n t on
L3
8
lie in a
s-I
Take for i n s t a n c e
L2, L3,
points
their H i l b e r t
lie in a h y p e r p l a n e
curve"
on
is
points
(n-h) (d-l)+2
normal
too.
of Theorem
be d i s t i n c t
' Ps denote
We note t h a t c o n d i t i o n s
conditions LI,
s
points
c a n be c o m p l e t e d
o n a rational
consequence
, and
2.3(1)
say
"'"
in w h i c h c a s e these
REMARK
(b)
Pl'
{b.} . z z ~ o
(i)
either
is a n i m m e d i a t e
L
points in such
function of l.ll(c)),
in
302
We n o w t u r n to the p r o m i s e d F i r s t recall called a that
divisor on
is s u b c a n o n i c a l
curve in
~n
C
degrees
t
denote,
d-3
...
We shall n o w p r o v e a few r e s u l t s
PROBLEM
2.6
:
w h i c h is l i n e a r l y
Let
normal,
is complete.
Does then
fixed p o i n t s
?
Clearly classical curves.
C ~
~n
i.e.,
C
the above p r o b l e m has an a f f i r m a t i v e
result,
due to H. Weber,
provides
basic
LEMMA
fact.
2.7
irreducible
:
n gd
projective
be a simple
curve.
T h e n the
of
d
distinct points
points
of
series w i t h o u t divisor
D
. Also,
a plane
after recalling assume
,
without
for n o n - s i n g u l a r
we shall
d
sections n gd
series
t = 1
section,
"general"
(cf. also [Se]).
curve o f d e g r e e
answer when
result,
linear
hypersurfaces
t = rl+...+rn_l-n-i
linear
of W e b e r ' s
( For the rest of this
Let
n-i
answer
C
irreducible
system of hyperplane
a positive
We n e x t give a slight g e n e r a l i z a t i o n
well-known
of
o f type
complete
such
plane curve of degree
be a s u b c a n o n i c a l
simple
is
s e c t i o n of
to the f o l l o w i n g p r o b l e m
such that the linear
admit a unique
t > 1
a non-singular
is subcanonical
(n ~ 2)
(n > 2)
a hyperplane
intersection
related
pn
an i n t e g e r
any non-singular
. M o r e generally,
, rn_ I,
curves.
C~---~
respectively,
w h i c h is the c o m p l e t e rl,
curve
if there exists
. For example,
o f type
(n > 3)
of respective
H, K
to s u b c a n o n i c a l
irreducible
curve o f type
I K I , where
and a c a n o n i c a l d > 4
that a non-singular
subcanonical
I tH I =
applications
a
char k = o . )
fixed p o i n t s o n an n gd
of
satisfies
the f o l l o w i n g (i)
D
consists
(2)
any
THEOREM G
n
2.8
:
Let
C
d e n o t e a simple c o m p l e t e
h > 1
and (a) (b)
m < d+l G G
is a
D
impose
independent
be a n o n - s i n g u l a r linear
, and
series
h gm
conditions
without
, each o f w h o s e d i v i s o r s
is the c o m p l e t e
2 gd
cut out on
gd
p l a n e curve of degree fixed p o i n t s
. T h e n o n l y the f o l l o w i n g p o s s i b i l i t i e s 1 gd-i
n
on
C
consists
for of
G d-i
b y all lines o f
d > on
C
5
and let , with
e a n occur
:
collinear
points
~2
,
303
Proof
:
we c a n w r i t e denote
their Hilbert
(i) in c a s e
cases
(2)
function
consists
• This
in the
2 gd+l
contradiction
d
and degree
d
points,
clearly
, hence
in v i e w
of Lemma
and let
is s p e c i a l ;
m ~ d-i
2.7
,
{b } i i ~ o
hence we get
. We n o w d i s c u s s
(cf. T h e o r e m
2.3(1))
we a r e
t
degree. which
:
Let
C
admits
Proof
:
Let
answer
Indeed,
if
.
d - i ~< b d _ 3 < d
b d _ 3 = d)
(resp. w i t h
to P r o b l e m
C
our
two
series
fixed points) :
2.6 in the c a s e o f a
is a s u b c a n o n i c a l
normal,
then
D = P1 + on
""" + P d c
Lemma
Suppose
2.1
not.
and the remaining
o f the d i v i s o r
is a c o n t r a d i c t i o n
n gd
our
without
n gd
{bi}
curve
in
~n
(cf. [ A - S ] ) we h a v e
:
b2
~<
cut out on
. So w e a r e d o n e
C .
divisor
2n ,
L
with
L
of
type
t
(I>2)
.
of a simple
denote hence
d-i
bt~<
o f the
. Hence, tD
linear
the Hilbert
. N o w we w i s h
is o u t s i d e by
(n i> 3)
fixed points
i i> o
2.4
1~n .
is s p e c i a l ,
by Corollary point
in
normal)
be a g e n e r a l
, and let
, we g e t
Then,
curve
is l i n e a r l y
simple
By Riemann-Roch,
and
L
.
o f the t h e o r e m .
is l i n e a r l y
(which
a unique
fixed points
are coplanar.
(resp.
(in
contradiction
(b)
, hence
fixed point
be a s u b c a n o n i c a l
d ~< (t+l) ( n - l ) + 3
2.7
in c a s e
bd_ 3 > d-2
with one
the proof
fixed point:
we are
b d _ 3 = d-i
gives an affirmative "low"
i.e.,
; otherwise
+ 2
2.9
of these points.
)
if
bd_ 3 = d-2
with one
implies
2.3(1),
1 a gd+l
(resp.
C
multiple
collinear
, which
Then
without
1 gd
of
and type
THEOREM
hyperplane
series
we have necessarily be a
curve of
(t+l) (n-l)
by Lemma
case,
. This completes
subcanonical
n gd
our
, hence
bd_ 3 = d-2
series would
Our ne~t result
I>
. Then,
a r e all d i s t i n c t ,
d-2 ~ m-h ~ m-i
implies
light of Theorem
w o u l d be a
d
P~ s l
our
m = d+l
of degree
the
. By Riemann-Roch,
that
. In t h i s
(i) above)
(3)
where
m
h gm
of our
. m = d
D
"'" + P
divisor
:
m = d-i
(a)
view of
be a general
. It f o l l o w s
all p o s s i b l e
Now,
D
D = P1 +
bd_ 3 = m-h
Hence
Let
series
function
d - n ~< tn
. Hence,
to s h o w t h a t the d
points
if we c o m p a r e
, we get a
1 gt
on
P"I s
lie on a the C
t, which
304
RI~4ARK 2 . 1 0 canonical C
curve
:
We recall
in
n
is a C a s t e l n u o v o
degree. proved
Hence,
(n ~
curve,
in p a r t i c u l a r ,
[A ]
3)
C
THEOREM the complete m = 3,4
:
Let
and C
admits
Proof
:
Let
these
points.
(I) of type sarily
m
=
n-i :
on
3
, we g e t
and also
lies on both
that
m n
= 4n-i
=
4
general
that Theorem
lying on a quadric two c u b i c
surfaces
intersection
irreducible
curve
to its
2.9 c o v e r s ,
surface
in
~3
3
in
~3
~3
degrees
, then
result
in
in
sub-
respect
which
m,n
. Then,
if
without
fixed points
be a general
divisor
of a simple
{bi}
i > o
since
ID I
. Now,
S
of
T
D
suppose
is
with
bn_ 1 >
4n-6
Theorem
. Hence
sequence
and
of
arguing
i ~ o
. Hence,
and
plane
i ~ o
to s h o w t h a t
bI = 3
, l.e.,
PI'
"'"
' P4n
neces-
is s t i l l a n
by Theorem conic,
C
2.3(2)
say
F .
is s u b c a n o n i c a l first observe
is s t i l l
(I) a b o v e ,
an
, hence we g e t a
contradiction. We w i s h
of
, and we are done
bn_ 1 = 4n-5
as in
function
is s u b c a n o n i c a l
. In fact,
{bi}
necessarily
C
{b.} 1
bI = 3
bn_ 1 = 4n-6
, we h a v e 2.3(2)
of
I D I is s p e c i a l
3 gmn
complete
. Then we have
lie o n a n o n - s i n g u l a r
since
(since the d i f f e r e n c e
and
2.3(1))
So
.
:
bI = 4
sequence
: contradiction.
b n = 4n-3
the H i l b e r t
is s p e c i a l
(in v i e w of T h e o r e m
points and
denote
the two cases
(since the d i f f e r e n c e
3n
applying
3
gm_n
separately
. In t h i s case,
, we g e t
Now,
, and let
= 3n-i
n
o f the
bn_ 1 > 4n-6
sequence).
from a more
of respective
complete
mn
bn_ 1 = 3n-3
b
Hence
(II)
"'" + P
. In t h i s case,
2n+2
F
simple
T h e n we d i s c u s s
at least
of type
C
b n _ I- b n _ 2 = 2
O-sequence)
S,T
(t+l) ( n - l ) + 2
genus with
observing
of
be a non-singular
a unique
D = P1 +
fixed points mn
intersection
d =
normal
n ~ m
Then
without
follows
intersection
o f two s u r f a c e s
is a l i n e a r l y
of maximal
in the c a s e o f a c o m p l e t e
C
intersection
C
and degree
it is w o r t h
the c a s e o f a c o m p l e t e
2.11
t
2.9 a l s o
. Furthermore,
say a bit more
t h a t if
is a c u r v e
Theorem
a n d a l s o t h e c a s e o f the c o m p l e t e
We c a n
[A-S ]
of type
i.e.,
in t h i s case,
by Accola
from
lie on a plane
.
Obn+ 1 =
305
S u p p o s e not. T h e n
bI = 4
, w h i c h implies
impose i n d e p e n d e n t c o n d i t i o n s to i n d e p e n d e n t c o n d i t i o n s to points
P4n-6'
I0~3(n-2)I
"'" ' P4n
:
At least
2n+3
Proof
(of the Claim) ~3
:
and let
P3' P4'
P4'
''" ' P4n-7
"'" ' P4n impose
o f the
7
P3
4n
points
of
W e first o b s e r v e that
D
lie o n a p l a n e
P4n-6'
. Otherwise, b y our hypotheses,
are in general position. Then, if we r e p l a c e
P4n-6
(see the p r o o f o f T h e o r e m 2.3(1)), we get that the are also in general position,
. Let
. T h e n e v e r y q u a d r i c surface through the
also contains
CLAIM
general p o s i t i o n in
I 0~3(n)I
bn_ 2 = 4n-9
"'" ' P4n
9
are not in
the p o i n t s
b y a suitable points
and their H i l b e r t function is
hence, b y L e m m a 2.2, t h e y lie on a r a t i o n a l normal curve,
°
P3' P4n-6''''' P'l
(4 ~ i < 4n-7)
P3' Pi' P4n-6' 1
say
4 E
P4n
7
9
"'" ' P4n ÷
;
. Now, b y r e p e a t i n g
an a r g u m e n t similar to the one d e v e l o p e d in the p r o o f o f T h e o r e m 2.3(1), it follows that at l e a s t S
and
T
3n+2
o f the
: contradiction
O n the o t h e r hand, o f the set
U = {P4n-6'
4n
points of
D
lie o n
7
÷
(2)
1
4
6
7
÷
(3)
1
3
5
7
÷
(4)
1
3
6
7
+
We n o w discuss all p o s s i b i l i t i e s
L
contains e x a c t l y
4
by
Pi
2n+5
1.7, we get (ib)
2n
points o f bn_ 1 >
U
o f the D
4n
P3 6 L 1
5
L
say
P4n-6'''''
and s p a n n i n g a plane
. Now, if we replace Pi E L 1
points o f
D
P4n-3 ' l y i n g o n L1
. Then,
it
(in the usual way)
; hence, i t e r a t i n g the argument, we lie on
L1
. Indeed,
L1
contains
; otherwise, b y L e m m a 2.7(2), T h e o r e m 2.3(1) and L e m m a
(2n-l)+(2n-4)
contains
U
c o p l a n a r points,
(4 ~ i ~ 4n-7), we have
get that at l e a s t at l e a s t
for the set
w i t h the r e m a i n i n g p o i n t s o u t s i d e
follows from our h y p o t h e s e s that P4n-2
lies o n b o t h
m u s t be one o f the following
"'" ' P4n } 4
a plane
E
in v i e w o f L e m m a 2.7 and T h e o r e m 2.3, the Hilbert function
1
U
; hence
.
(i)
(la)
E
= 4n-5
: contradiction, w h e n c e the Claim.
points o n a n o n - s i n g u l a r conic
F
lying in a p l a n e
L
,
306
with
the
remaining
sibility
P4n-6'''''
U
P4n-i
replacing
P4n-6
that at least
back
coplanar
points
M >
. Also,
2n+3
lying,
say,
2.3(1), Hence {d.} l
let
we m a y
contradicts
could
answer
seen
that
this pos-
of
and not lying
. Now,
through
and
D
we m a y
assume
L
P4n-i
concludes
loss . Then,
the argument,
, which
say
(without
P4n-5'''''
iterating
lie on
on a conic,
we g e t the
. can be made
in e a c h o f t h e o t h e r
of the theorem,
{Pl''''' . • > o
, which an
cases
(2),
S N L
and
d i = 4i-2
contain,
3n
,
(3),
(4),
of
these
(cf. L e m m a
of
we have M
:
points
2.7(2), = 4n-3
the d i f f e r e n c e
sequence
M = d +3 > n
Theorem
of
3n+3 plane
i = 4,...,
b I = 4 , made
above,
number
b n = dn+(4n-M)
(since
for
quartic.
n-i
Then we
. Hence
M
=
easily 4n
at the beginning.
of the theorem.
that
the a r g u m e n t
the U n i f o r m
c a n be r e a r r a n g e d
for a n y c o m p l e t e
way
, we g e t
is an i r r e d u c i b l e
C. C i l i b e r t o
, in a d i f f e r e n t
~
function
that
by using
2.6
2n-3
the maximum
b y the C l a i m
the H i l b e r t
= 4n-6 n
denote
Hence
the p r o o f
above
d
M
; then,
4n-M <
implies
let
O-sequence).
observing
to P r o b l e m
P4n }
denote
. Since
the a s s u m p t i o n
argument
will
P3 E L
(as usual)
bn_ 1 = d n_l+(4n-M)
completes
Finally,
L
lie o n the c o n i c
points
(i)
L
assume
be s h o r t e n e d ,
t h a t the
which
{d.} l
d 3 = iO
It is w o r t h
on a plane
P. l
4n
to t h e p r o o f
is s t i l l
which
it is e a s i l y
.
1.7):
d2 = 6 ,
This
the
in the s e t
get:
. Then,
. Hence
not
discussion
d n- dn_ 1 = 3 i > o
does
in c a s e
on a plane
Lemma
Now,
2.11
of
a similar
L
points
P4n ~ L
P3
2n+3
so we a r e d o n e
Going
6
by a suitable
o f the C l a i m Now,
of
' with
that
outside
out.
consists
of generality)
and
points
can be ruled
(ic)
proof
2
informed
Position
in t h e p r o o f
Lemma
these
general results.
to p r o v i d e
curve
me t h a t h e p r o v e d
of T h e o r e m
(cf.[H 1 ]).
in s u c h a w a y
intersection
(and i n a m o r e
in p a r t i c u l a r ,
given
in
both
context)
~3
Also,
2.11
we hope
an affirmative .
Theorem
2.9 a n d T h e o r e m
: he is w r i t i n g
a paper
,
307
3.
OPEN PROBLEMS AND FINAL REMARKS
Let
S
be the H i l b e r t
then it is c l e a r Hilbert
that there exist,
function
geometric
f u n c t i o n o f a finite set o f p o i n t s
S
in general,
. So, a first naive p r o b l e m
properties
?" . Such a q u e s t i o n
previous
sections
is d o u b t l e s s
some e x p r e s s i v e
results
a n d R e m a r k l.ll(c)
. In this connection,
of points
is to c h a r a c t e r i z e
~n
to give a c o n v e r s e the H i l b e r t
o f the B ~ z o u t
to g e t the d e s i r e d Now, approach
result
intersection. Remark
let us start w i t h the c a s e
n = 2
is to appeal
consisting Bacharach of
Z
of
mn
d i s t i n c t points,
property,
PROBLEM points,
with (a)
Z Z
:
Let
degrees
Z
sufficient
that
when Z
have
is n o t enough
T h e n o n e w a y we can
m+n-3
Z
2
in
, is said to h a v e the C a y l e y passing
in
p2
through
consisting
(with no c o m m o n components)
intersection
intersection.
that,
(i.e.,
r e s u l t d u e to C a y l e y and B a c h a r a c h
f u n c t i o n o f the c o m p l e t e
all b u t one p o i n t
of
mn
distinct
However,
m = n ~ 3
property
of two curves
, and
.
(b) alone does n o t i m p l y n e c e s s a r i l y
it is shown in [ G - H ]2
, the C a y l e y - B a c h a r a c h
be a c o m p l e t e
intersection
?
It is fairly clear t h a t c o n d i t i o n
context)
n
in q u e s t i o n
this a s s u m p t i o n
, for simplicity.
3 < m < n
be a z e r o - c y c l e
has the C a y l e y - B a c h a r a c h
is a c o m p l e t e
that the c y c l e s
in
set
, such t h a t
m,n
a complete
zero-cycles
2.3
Z
h a s the H i l b e r t
of respective
Is then
contains
3 < m < n
Z
(b)
3.1
with
Theorem
a b o u t a finite
following [ G-H ] 1 ' that a zero-cycle
if a n y curve o f d e g r e e
necessarily
saw in the
2.5(b)).
to a c l a s s i c a l
(cf. [ S - R ] , [ G - H ] 1 ) . F i r s t recall,
;
"Are there a n y
in particular,
question
Clearly
2)
having a given Hilbert
intersection
, assuming
(n >
sets of p o i n t s w i t h
y e t we a l r e a d y
a v e r y natural
(cf., e.g.,
the p r o b l e m
~n
in this direction:
theorem)
function of a complete
in
too vague,
complete
~n
is the following:
s h a r e d b y all sets o f p o i n t s
function
in
quite d i f f e r e n t
in
intersection.
(and in a m o r e
property
that general
is "in g e n e r a l "
O n the o t h e r hand,
if we assume
Z
308
moreover
that
Bacharach We prove
Z
lies on an irreducible
property feel
that Problem
that, o n l y
in
conditions
3.2
2
(a),
:
(b)
degrees
component.
Now,
(i) and
and
G
and
points:
1
the
; hence,
F
and
points
of
Finally,
Z
3n
exactly the
set
their
line
one
L
(cf. a l s o [ S ~.
we a r e a b l e
(n >
component); ,
above, 3)
intersection
to
that
G
does
and
in c o m m o n
of a plane
be a
cubic
hence
n
of
Z
and
F
of
as a the
F = LF'
lying on the conic from
points
<
contains
F, G,
; so w e c a n w r i t e
n+2 c
say
Then we consider
It f o l l o w s
at l e a s t
Z'
Z
satisfying
not contain
intersection.
L
let
.
function.
contains
2.3(1)
3.1
, of all points
Hilbert
have one conic
also
F
C
in t h i s case,
out that Problem
t h a t we w o u l d
Hilbert
in c o m m o n
would
Another
number
line
, z'
Cayley-Bacharach
the minimal
Yet,
l i e s o n two c u r v e s ,
is n o t a c o m p l e t e
by Theorem
we point
a given
Z
in
points
components
the property
the g e n e r a l i z e d
between
distinct
, our cycle
(otherwise
problem
intersection
2n-2
(b) t h a t
of
Z
. This
at least
F'
n+l
(otherwise implies
that
collinear
.
G Z
, then the Cayley-
answer.
introduced
is the c o m p l e t e
have a common
we g e t a c o n t r a d i c t i o n
using
Z
denote
contradiction (2)
2n+2
would
of
with
have
. Also,
G'
Cn_ 1 = Cn-
i ~ o
1
the notation
(a)
. Now consider
{ci}
Cn_ 1 < CnF'
:
G = LG'
and let
suppose
m
is a c o m p l e t e
an a f f i r m a t i v e
, with no common
3, n,
cases
F
With
In v i e w o f
Z
of degree
m = 3
. Then
n
respective
two p o s s i b l e
:
that
3.1 a d m i t s
consisting
a curve of degree
Proof
implies
in t h e c a s e
PROPOSITION zero-cycle
certainly
curve
function
of g e n e r a t o r s
S
be a c o m p o n e n t which
C
G ). T h e n , the proof
here
in
in [ G - H ]
regards
set of points
for the homogeneous
contains
stated also
defined
to d i s c u s s
of a f i n i t e
of
concludes
3.1 c a n b e
property
like
. Hence
in
at least using
(b)
,
.
~n
, with
n ~
2
the r e l a t i o n s h i p ~n
(n >
ideal of a set of points
2)
and
with
3 ,
309
Hilbert
function
S
we w i l l
s t i c k to the " g e n e r i c "
DEFINITION points PI'
in
3.3
~n(k)
"'" ' P
. I n s t e a d o f t r y i n g to s t a t e the p r o b l e m corresponding
(s > n > 2)
b. l
with Hilbert
position
=
min
((i+n) n
, s)
the f i r s t b a s i c p r o p e r t i e s
PROPOSITION and let
I
=
points
in
of
. Then
of
I
3.4
...
(a)
the ideal
(b)
~(I)
we h a v e (i)
=
, X
I di~
generated
(c)
Let
I d • Id+ 1 O
k [Xo,
Id+ 1
:
by
i f we w r i t e
n
PI'
"'"
] . Also,
is g e n e r a t e d
;
s =
((d-l)+n) n
s-position)
if
i ~ o
in
n
( c o n s i d e r e d as p o i n t s ( ~n) s
o f a set o f p o i n t s
in g e n e r i c
position
be points
in
~n(k)
9(I)
denote
+ h
, where
W
d i> 2
and
, with
ideal o f t h e s e
the m i n i m a l
b y f o r m s o f d e g r e e ~< d + l
.
in g e n e r i c p o s i t i o n
, b e the h o m o g e n e o u s
I d + d i m k Id+ 1 - d i m k W Id
. Then we say that
o p e n set o f
Id # O let
points
be distinct
form a non-empty
' Ps
... , w i t h
Ps
{b.} l i ) o
for a n y
s
definition.
P1 . . . . .
(or a l s o in g e n e r i c
( ~n) s ) w h i c h a r e in g e n e r i c p o s i t i o n
N e x t we r e c a l l
Let
function
It is s h o w n in [ G - O ] l t h a t the sets o f in
to the f o l l o w i n g
([G-O ]I' [ G - O ] 2' [ G - M ] 2 ) :
a r e in g e n e r i c
s
case,
in its full g e n e r a l i t y ,
number of generators
; denotes
the
k-subspace
o ~< h ~< (
(d-i) +n) - 1 n-i
~(I)
(~+~)-(n-l)
,
: [ (d+n-l) n-i
Proof
:
_
h ] - m i n {n [
(a) f o l l o w s
Cohen-Macaulay
standard
consequence
(a)
(i)'
of
(d+n-l)_
directly
G-algebra
. Also,
n-i
(cf., e.g.,
(n-l)
(b)
~
We s h a l l n o w s h o w the f i r s t i n e q u a l i t y (the g e n e r a l
argument
So, l e t
... , F m
FI,
(d+n) n-i
from some general
in v i e w o f
2 d i m k Id +
h ]-
,
di~
in
Id
<
properties
[ G-M ] 2 )
and
W
~
while
of a
1-dimensional
to
min {(n+l)di~
(i)' in the c a s e
<
(b) is an i m m e d i a t e
(i) is e q u i v a l e n t
b e i n g c l e a r a f t e r that), be a b a s i s o f
, O }
n = 2
the o t h e r
I d , d i m k Id+ I}
, for s i m p l i c i t y
inequality
(over k ). T h e n w e m a y a s s u m e
is i m m e d i a t e . (without loss
310
o f generality) question
that the c o - o r d i n a t e
and f u r t h e r m o r e
are l i n e a r l y
axes of
forms
c o n t a i n none o f the
F1 . Hence,
[0:O:i ] ~
independent
2
~
(over k ) in
W
s
points
XoF1,...,XoF m, XlF1 , . . . , x l ~ , , a n d so we are done
in
X2F1
.
N o w we are able to state o u r problem.
PROBLEM d ~
2
3.5
n ~ 2
:
Let
and
s
o < h ~
'
satisfying generic
the i n e q u a l i t i e s
position
REMARK Zariski in
:
for w h i c h is equal
conjecture
n ~
conjecture
( n)s
3
[ G - M ]2
(see [ G - M ]i
given in P r o p o s i t i o n
s
n = 2
in
(see also [ G ]
points
in
2
, a set of
~ (i(Pl, " .., p
s-tuples
9(I)
of p o i n t s
, with r
s
points
))
s
=
r
?
a non-empty
in g e n e r i c
position homogeneous
in P r o p o s i t i o n
. This
3.4(c)
(positive)
results
we p o i n t o u t that in o r d e r to p r o v e
are this
, it w o u l d be e n o u g h to produce,
position
in
. Further,
n
for the m i n i m a l d
for
whose homogeneous
the u p p e r b o u n d
s h o w n in
the c l a s s i c a l
such that
in
(of the c o r r e s p o n d i n g
, while o n l y a few
with
, [D-G-M ] )
+ h
for any i n t e g e r
t h a t there exists
similar bounds
, it c o i n c i d e s
[ G-O ] 2
for and
ideal
9(I) [G ]
;
upper bound given by Dubreil
n u m b e r of g e n e r a t o r s
o f the
is the l e a s t d e g r e e o f a c u r v e
.
3.5 admits
number of generators
c a n take all p o s s i b l e
moreover
of
in generic
improves
Next we show that Problem
2
[ G - O ]2
) . Also,
ideal o f a finite set o f p o i n t s
in
(i) a b o v e
such t h a t
s
an o p e n condition)
3.4 (c)
when
the m i n i m a l
P
number of generators
moreover,
them
given in
number of generators
in
((d-~)+n)
Does there exist,
in
consisting
s =
"
to the lower b o u n d g i v e n for
, a set o f
containing
9(I)) PI'''''
the m i n i m a l
has the r e q u i r e d m i n i m a l
[D ]
(for
It was c o n j e c t u r e d
(which e x p r e s s e s
s > n
((d-l)+n. 1 n-i )-
, say
has b e e n p r o v e d
known w h e n
any
3.6
pn
o p e n set in
n
ideal)
in
be an integer o f the f o r m
such p o i n t s
this is no l o n g e r
an a f f i r m a t i v e
o f the ideal o f a set o f p o i n t s
values
(allowed b y P r o p o s i t i o n
are in uniform position in
true
(see [ G - M ]2
We first state a simple
answer when
lemma.
)
2
([ G - O ]i
n = 2
in generic
3.4(c))
, i.e., position
. However,
' [ G - O ]2
' [ HI ])
if '
311
LEMMA
3.7
1 ~ q < m+l lower
given
exist
have
be an i n t e g e r
t' = t-(m+2)
in
any
t
~ ( I ( Q 1 ..... (2)
t
for the m i n i m a l position
(i) there
Let
, and let
bound
in g e n e r i c
:
2
=
the
. Also,
form
let
9o,t
of generators
m+2 ( 2 ) + q
t =
(resp.
, with
9o,t,)
o f the i d e a l
of
m ~
denote
t
2
,
the
(resp.
t') p o i n t s
, say
PI'''''
. Then
set o f
points
Qt ) )
number
of
t'
points
in g e n e r i c
in g e n e r i c
position
in
~
position
2
in
, say
p2
QI'''"
Qt
such
Pt'
that
~(I(Pl, .... P t , ) ) + l
i f we w r i t e
u = [ (m+2)-
~o,t ]+i
,
u' = [ (m+l)-
~o,t'
]+i
, t h e n we
u' ~ u < u ' + l
Proof
:
(1)
Let
RI,...,
Rm+ 2
denote
~,...,
Rm+ 2
are
L
any
be a line m+2
in g e n e r i c
in
distinct position
p2
missing
points in
on
p2
L
PI''''' . Then
. Now,
Pt'
the
applying
and
t
let
points
Proposition
~'''''Pt' 3.4(b)
,
we a r e d o n e . (2)
This
THEOREM d >
2
and
follows
3.8
:
essentially
Let
o ~ h < d
s
exist
s
points
statement
be an i n t e g e r
. Then,
given
(d+l-h)
there
from
of the
{d-2h
position
, O}
in
~
~ ( I ( P l , . . . , Ps ))
Proof immediate. i t for
the
light (b)
Then,
by
to be s h o w n ) . (and w i t h
9o,s
~o,s'
of Lemma
3.7(1)
by using
=
, with Now,
~o,s'
induction
+ 1
true
write
3
with
~
r
2
~
introduced case
the proof.
((d-l)+2. 2 ~ + h
, with
,
PI'''''
such
Ps
that
r
d ~
for a n y i n t e g e r and
concludes
d+l
, say
=
=
(~) +
there), the
2
, the
< d-i
1 ~ h < d
s' = s-(d+l)
• in w h i c h
s =
r
on the integer
d >
the n o t a t i o n
=
~o,s
induction
the result
(d-l)+2 ( 2 ) + h
3.7(2) (a)
We p r o c e e d
So, we a s s u m e
s =
is n o t h i n g Lemma
:
, which
form
any integer
- min
in g e n e r i c
(1)
we have
theorem
d = 2
and we wish
(clearly, (h-l)
case
if
only
follows
to p r o v e
h = o
. Then,
being
in v i e w
, there of
two p o s s i b l e by
cases:
induction,
in
; " NOW, and Lemma
the
first
3.7(1)
case
again,
r = 9o,s we a r e d o n e
is s e t t l e d .
in
[G-M
]
'
312
Finally, complete
we w o u l d
that a
S t a n l e y r a i s e d the f o l l o w i n g
function o f a c o m p l e t e
R
actually
2-dimensional
reduced
EXAMPLE
3.9
distinct
lines
in
(all d i s t i n c t
points on
in
Let 2
d > 1
from ,
standard
o f two p l a n e
be any positive
0 )
2i-i
curves of d e g r e e
number o f g e n e r a t o r s
through
points on
d
can we c o n c l u d e
t h a t has the H i l b e r t (cf. also [ G - M - R ])
(Cohen-Macaulay)
intersection
integer
a point
L i , ...
standard
and n e v e r t h e l e s s
,
O
a n d let
L l,
...
. T h e n choose take
1
2d-i
p o i n t on
points on
components
ideal o f these p o i n t s
type o f the h o m o g e n e o u s
. Also, is
, Ld d2
Ld
f u n c t i o n o f the c o m p l e t e
, w i t h no c o m m o n
o f the h o m o g e n e o u s
d
has the
he gave an e x a m p l e o f
ring
reduced
in the f o l l o w i n g way:
e.g., [ G - O ]2 )' the C o h e n - M a c a u l a y is p r e c i s e l y
G-algebra
R
type.
e a s y to c h e c k that these p o i n t s h a v e the H i l b e r t
points
function of a
G-algebra
. Furthermore,
f u n c t i o n of a c o m p l e t e
, all p a s s i n g
L 2 , ...
?"
1-dimensional
Cohen-Macaulay
:
: "If a
r i n g a n d yet is not a G o r e n s t e i n
t h a t h a v e the H i l b e r t
have any prescribed
3
Cohen-Macaulay
question
under w h a t c i r c u m s t a n c e s
intersection
n o w s h o w t h a t there exist
G-algebras
~2
intersection,
is a c o m p l e t e
function of a Gorenstein We shall
about the H i l b e r t
intersection.
In [ St ], Hilbert
like to m a k e a b r i e f c o m m e n t
be
points L1 , . It is
intersection
the minimal
d+l
co-ordinate
, hence
(cf.,
r i n g of these
313
REFERENCES
[A]
R.D.M. Accola, On Castelnuovo's inequality for algebraic curves, I , Trans. Amer. Math. Soc. 251 (1979), 357-373.
[A-S ]
E. Arbarello and E. Sernesi, Petri's approach to the study of the ideal associated to a special divisor, Inventiones math. 49 (1978), 99-119.
[D]
P. Dubreil, Sur quelques propri~t4s des syst~mes de points dans le plan et des courbes gauches alg4briques, Bull. Soc. Math. France 61 (1933), 258-283.
[ D-G-M ]
E.D. Davis, A.V. Geramita and P. Maroscia, Perfect homogeneous ideals of height 2 in polynomial rings: Dubreil's theorems revisited, I, (to appear).
[G]
A.V. Geramita, Remarks on the number of generators of some homogeneous ideals, Bull. Soc. Math. France (to appear).
[G-H ]i
P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978.
[G-S l
, Residues and zero-cycles on algebraic
2
varieties, Annals of Math. 108 (1978), 461-505.
[G-K ]
C. Greene and D.J. Kleitman, Proof techniques in the theory of finite sets, Studies in Mathematics, Vol. 17, Mathematical Association of America, 1978, 22-79.
[G-M ]1
A.V. Geramita and P. Maroscia, The ideal of forms vanishing at a finite set of points in
pn
, C.R. Math. Rep. Acad. Sci. Canada 4 (1982),
179-184.
[G-M ]
, The ideal of forms vanishing at a finite
2 set of points in
~n
, Queen's University Mathematical Preprint
No. 1981-5 .
[ G-M-R ]
A.V. Geramita, P. Maroscia and L.G. Roberts, The Hilbert function of a reduced
K-algebra, Queen's Papers in Pure and Applied Mathematics,
No. 61 , Kingston, Ontario, 1982, pp. CI-C63 .
314
[G-O ] l
A.V. Geramita and F. Orecchia, On the Cohen-Macaulay type of n+l in A , J. Algebra 70 (1981), 116-140.
[G-O ]2
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, Minimally generating ideals defining certain tangent cones, J. Algebra 78 (1982), 36-57.
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J. Harris, The genus of space curves, Math. Ann. 249 (1980), 191-204. , A bound on the geometric genus of projective varieties, Ann Scuola Norm. Sup. Pisa
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8
(1981), 35-68.
F.S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531-555.
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P. Maroscia, The Hilbert function of a finite set of points in
~n
,
Queen's University Mathematical Preprint No. 1982-14.
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L.G. Roberts, The Hilbert function of some reduced graded
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B. Segre, Sui teoremi di B~zout, Jacobi e Reiss, Ann. Mat. Pura Appl. (4) 26 (1947), 1-26.
S-R ]
J.G. Semple and L. Roth, Introduction to algebraic geometry, Clarendon Press, Oxford, 1949.
[Se ]
E. Sernesi, On the problem of uniqueness for certain linear series (talk given at the Conference).
[St ]
R.P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), 57-82.
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B. Teissier, Vari~t4s toriques et polytopes, S4m. Bourbaki 1980/81 , Lecture Notes in Mathematics, Vol. 901, Springer, Berlin 1981, 71-84.
P.S. After this paper was typed, an affirmative answer has been given to Problem 3.1 ; the proof will appear in
[ D-G-M ]
" H o m o g e n e o u s Bundles
V.B.
Mehta
in c h a r a c t e r i s t i c
and
p"
A. R a m a n a t h a n
Introduction
Let X = G/P.
G
be a reductive group,
In [4] R a m a n a n proved,
representation associated
bundle
stable bundles. parabolic
O : P ÷ GL(n), V
on
O
P
is maximal,
is trivial on the radical of
is semistable,
and hence are r e s t r i c t e d
we prove the result
the results hold
in c h a r a c t e r i s t i c
We make e s s e n t i a l
[7]
for a r b i t r a r y
theorems of
in c h a r a c t e r i s t i c
zero. p.
It is
In this note
p.
use of the notion of a H a r d e r - N a r a s i m h a n
for a n o n s e m i s t a b l e
semistability
P, the
in fact a direct sum of
to c h a r a c t e r i s t i z
p r i n c i p a l G - b u n d l e on
We also need the fact that on a h o m o g e n e o u s preserves
s u b g r o u p and that for any
Their m e t h o d s use the vanishing
natural to ask whether
filtration
which
a parabolic
This result was e x t e n d e d by U m e m u r a
subgroups.
Borel-Weil-Bott
X
when
P
and consequently,
X [See Thm. 3.1].
space the Frobenius m o r p h i s m
associated
bundles of semistable
bundles are again semistable.
We would like to thank S. R a m a n a n suggestions. R a m a n a n a l s o
has a d i f f e r e n t
and M. Nori
Professor
Naples and the C.N.R.
the first-named
at the U n i v e r s i t y of Naples,
N a z i o n a l e delle R i c e r c h e of Italy.
and
proof of our result.
During the p r e p a r a t i o n of this paper, a Visiting
for d i s c u s s i o n s
He is grateful
for their hospitality.
author was
supported by the C o n s i g l i o to the U n i v e r s i t y of
316
§ 2. In this section we do the p r e l i m i n a r i e s theorem.
We deal with the following
projective and
H
situation
v a r i e t y over an a l g e b r a i c a l l y
is an ample line bundle on
X.
n e e d e d for the main
: X is a n o n s i n g u l a r
closed
field
k
of char p > 0,
For the d e f i n i t i o n s of
H
s t a b i l i t y and s e m i - s t a b i l i t y and for the notion of the H a r d e r - N a r a s i m h a n filtration of an u n s e m i s t a b l e N o w let
Theorem if
~ : X ÷ X
2.1.
i)
0 = T0C
~r(T)
TIC
~ 0.
vector
bundle on
X, we refer to [2].
be the F r o b e n i u s morphism.
Let
T = T
Tr = T
Then we have
be the t a n g e n t bundle of
X
is the H a r d e r - N a r a s i m h a n
X.
Assume that
filtration of
Then for any semistable vector B u n d l e
V
on
X,
T,
then is
~(V)
also semistable.
2)
If
~r(T) ~
0, then,
for stable
V
on
X,
~
(V)
is also
stable.
Proof. V
We induct on the rank of
of rank less than
n,
~
[5, Thm. 3.23] that whenever than
n
,
that
and semistable. - subbundle of TX ÷ Hom(B,
VI~V If
2 ~
V1
for
i).
is semistable. and V 2
Suppose Then
for all s e m i s t a b l e it follows
are semistable,
is also semistable. (V) is not semistable,
~ (V).
~ (V)/B).
(V)
V
N o w let B C~
We get a c a n o n i c a l m a p
f :
By induction we get that Hom(B,
by s e m i s t a b l e
bundles of n e g a t i v e degree.
follows that
f
must be zero.
each of rank less V
let
from
be of rank (V)
~ (V)/B)
By our a s s u m p t i o n on
n
be the
is filtered TX
it
317
By purely inseparable descent theory it follows that
B
subsheaf of
V.
V,
is semistable. on
X.
contradicting the semi-stability of Now assume that
Suppose
W~
g
(V)
with
~r(T)
> 0
~(W)
=
and let ~g
(V).
'(V)/W are semistable and again we get a map By l) W
W ~
~ (V)/W
V
V,
Hence
~ (V)
be a stable sheaf
Now b o t h
W
f : T x *Hom(W,
is semistable, and of degree zero.
descends to a subsheaf of
descends to a
and
~ (V)/W).
Hence f=0, and
contradicting the stability of
V. Q.E.D.
Remark 2.2.
If
TX
then we always have
is generated by ~r(T) ~ 0.
H°(X, TX)
(e.g. homogeneous spaces)
Hence the above Theorem is a slight
improvement of Prop. l.l in [3].
For example, let and if Then
X TX
is defined by
X
be a complete intersection in ]pn , dim X > 3, t (fl'''ft)' we assume that ~ deg fi -< n+l i=l
is semistable of nonnegative degree, although
o
TX
may not have any
sections.
Remark 2.3.
Applications of the above to uniformization of semistable
bundles on simply-connected varieties and other related questions will appear separately.
There is also a partial converse, which enables one to
construct bundles whose Frobenius pull-back is not semistable.
318
§3. Let
G
be a reductive
and a Borel
subgroup
We call
a stable
E
of structure
T.
(resp.
g r o u p of
has c o d i m e n s i o n character
B~
g r o u p over
k.
~t
be a p r i n c i p a l G - b u n d l e over
E + X
semistable)
E, r e s t r i c t e d
We fix a m a x i m a l
G-bundle
T X.
if for any r e d u c t i o n
to any open subset whose c o m p l e m e n t
> 2, the line bundle a s s o c i a t e d
has degree
torus
< 0 (resp. ~ 0).
to any d o m i n a n t
See [6].
We then have the following result.
T h e o r e m 3.1. projective
Let
E + X
nonsingular
structure g r o u p of
i)
the
be a n o n s e m i s t a b l e
variety.
(U
semistable
2)
P + P/U
P) o b t a i n e d by
from the reduced P - b u n d l e
of
P
which can be e x p r e s s e d
p o s i t i v e c o m b i n a t i o n of simple roots the a s s o c i a t e d s t r i c t l y greater
than zero
Borel s u b g r o u p c o n t a i n e d
in P).
[4,7] on h o m o g e n e o u s
field has c h a r a c t e r i s t i c
Theorem
3.2.
unipotent ~bundle group
Let
r a d i c a l and obtained
P~
~
P
(simple roots taken with respect
to any
See [6].
bundles on
and
G/P to the case when the base
p.
be a p a r a b o l i c M
as a
line bundle has
We make use of this result to extend a t h e o r e m of R a m a n a n Umemura
is a
P/U - bundle.
for any n o n t r i v i a l character
degree
is a
s u b g r o u p such that
being the u n i p o t e n t radical of
the e x t e n s i o n of structure g r o u p
X
Then there is a unique reduction of
E to a proper p a r a b o l i c
P/U - bundle
G-bundle where
s u b g r o u p of
a Levicomponent
from the P-bundle
is semistable.
so that
G + G/P
G.
Let
P = M.U.
U
be its
Then the
by the e x t e n s i o n of structure
For any irreducible
r e p r e s e n t a t i o n of
M
319
the a s s o c i a t e d vector bundle on
is semistable.
Proof.
Iet
G ÷ G/P
by e x t e n s i o n of structure group.
bundle
i.e.
E + G/P
G/P
G
be the a s s o c i a t e d
acts on
with its action on
E
G/P.
P/U - bundle o b t a i n e d Then
it is a h o m o g e n e o u s
as a bundle a u t o m o r p h i s m g r o u p c o m p a t i b l e If
E ÷ G/P
is not semistable
then by
T h e o r e m 3.1 there will be a unique reduction of structure parabolic
subgroup
P'
of
Because of the u n i q u e n e s s action of
G
subgroup
P'
impossible
on
E.
M
P'
of this r e d u c t i o n
As is e a s i l y seen this implies
that the p a r a b o l i c
the adjoint action of
is a proper n o n - n o r m a l
2.2 and the result proved
as well as all its F r o b e n i u s bundle
G/P.
it is invariant under the
M
s u b g r o u p of
The last s t a t e m e n t of the theorem now follows Remark
g r o u p to a
over a suitable open subset of
is invariant under
since
from
which
is
M.
from T h e o r e m 2.1,
in [5, T h e o r e m 3.22]that
if a bundle
twists are s e m i s t a b l e
then any a s s o c i a t e d
if
g r o u p and
is semistable.
Remark 3.3.
More g e n e r a l l y
H
is a reductive
a h o m o m o r p h i s m such that the c o n n e c t e d c o m p o n e n t of the goes into the center of
H then the e x t e n d e d
M÷
H
center of
is M
H-bundle o b t a i n e d
from
E
is
in c h a r a c t e r i s t i c
zero also,
semistable.
Remark
3.4.
Note that our proof goes through
m a k i n g use of [5, T h e o r e m
3.18].
See also [i].
320
REFERENCES
1)
Kobayashi,
S.
:
Curvature
Proc. Japan Acad.
2)
Maruyama, Nagoya
3)
~.
~th.
~ehta, V.B.,
Set. A,
Boundedness
J.
78
(1980) ~V.
Ramanan, Topology,
5)
Ramanan, flag.
6)
7)
S. 5
:
bundles.
(1982). of semistable
sheaves of small ranks.
Semistable
sheaves on homogeneous
spaces
(preprint)
Holomorphic
vector
bundles
on homogeneous
spaces.
(1966).
S., Ramanathan,
A.
:
Some remarks
on the instability
(preprint)
Ramanathan,
A.
:
M~duli
Geometry
Proceedings,
Springer
1979.
Umemura,
H.
(1978),
of vector
65-94.
:
and abelian varieties.
4)
58
:
Nori,
and Stability
:
for principal
Copenhagen
1978.
bundles.
In : Algebraic
Lecture Notes
On a theorem of Ramanan.
Nagoya
732.
Math. J. 69
131-138.
A. Ramanathan School of ~ t h e m a t i c s Tata Institute of Fundamental Homi Bhabha Road Bombay 400 005 INDIA
Research
V.B. ~ h t a Dept. of ~ t h e m a t i c s University of Bombay Bombay 400 098 INDIA
The Group of Sections on a Rational Elliptic
Surface
by Ian Morrison*
(Department of Mathematics, U n i v e r s i t y of Toronto, Toronto, Canada) and
Ulf Persson**
(Institut Mittag-Leffler, Djursholm, Sweden)
The d i o p h a n t i n e
n
problem
= n. - i i=l l
(i) k 3n = [ n i + 1 i=l has for
k <8
only a finite number, of solutions. 2
n
Indeed
k
= [ i=l
(ni)2-i
k l([n~)2 z~ i=l ~
-1
= ~9 n 2 + 0 (i).
*Research p a r t i a l l y supported by the National Science an@ E n g i n e e r i n g R e s e a r c h Council of Canada * * R e s e a r c h p a r t i a l l y supported by the Swedish Research Council of Natural Sciences
322
As is w e l l - k n o w n , minimal
rational
however, elliptic
if
f:X
surface
, ~i
is a r e l a t i v e l y
then e v e r y
gives rise to a s o l u t i o n of these e q u a t i o n s c h o i c e of an o r i g i n X
section
S0 ,
into an a b e l i a n g r o u p u n d e r
addition.
If
X
is g e n e r i c
have i n f i n i t e order; many
solutions.
with
in a suitable
for
k = 9 ,
X
with
k = 9 .
numerically
Let
First,
the g r o u p
sense any
W h i l e we w e r e w r i t i n g
the e q u a t i o n s
Manin's
have
denote
enumerate ~
the paper,
solved by M a n i n solutions
o n the second.
will infinitely [ 6]
surfaces with infinitely curves
many
of the
[ 3 ] .
to a n s w e r
the set of s o l u t i o n s ~ .
Second,
to
(I)
describe
induced by t r a n s p o r t - o f - s t r u c t u r e . Igor D o l g a c h e v
out to us that the first of these p r o b l e m s had b e e n
S ~ SO
the sections.
~
law on
The
of f i b r e - b y - f i b r e
This paper g r e w out of the authors'~ a t t e m p t s problems.
X
was first used by N a g a t a
of r e l a t i v e l y m i n i m a l
are e x a c t l y
of
k = 9
c u r v e s of the first kind : the e x c e p t i o n a l
first kind on
two n a t u r a l
S
the set of s e c t i o n s of
the o p e r a t i o n
This c o n s t r u c t i o n
to give e x a m p l e s exceptional
hence
makes
section
Our m e t h o d
kindly pointed
and part of the second enables
us to s i m p l i f y
to the first p r o b l e m and to c o m p l e t e
his r e s u l t s
323
We first recall of curves
in the plane d e f i n e d
particular, through
each of nine points such systems
the base points of rational correspond
elliptic
all of whose
fibres
are
homogeneous
intersection
the c o r r e s p o n d i n g
with are
#(X)
the crucial
recover
Manin's
simplify the We
(any)
of
exceptional
and the c o m p u t a t i o n
points
on
X
#(X)
are the X
on
~
irreducible. the
formulae
(17)
by i d e n t i f y i n g formulae,
which
are then used to
~; in particular,
they g r e a t l y
the subgroup
of a b l o w d o w n The first
NS(X)
X
in terms of the
numerical
These
generated
by
~:X ---+ ~ 2 . is the d e t e r m i n a automorphisims
on
NS(X)
of the d i v i s o r s
The of
X .
We plan to take up in a subsequent generalizations
#(X),
These
of fibre p r e s e r v i n g
in
are themselves
we study the class
of its r e p r e s e n t a t i o n
is the i d e n t i f i c a t i o n
n-torsion
of
applications.
Aut0(X)
n i -fold
We then d e t e r m i n e
induced
divisors
in
the special
we then call
in our treatment,
enumeration
of the group
~ .
structure
irreducible
give two a d d i t i o n a l
second
of
(16), and obtain
novelty
passing
sections,
is well-known).
structure
n
Next,
whose
the key step of i d e n t i f y i n g
standard
tion X
for
X
irreducible;
form on X
conditions,
the nine points
of cubics.
space
systems
We illustrate
to elements
(Most of this m a t e r i a l principal
P. z
surfaces
bijectively
linear
of degree
have when
of apencil
about
by base point
the system of curves
properties
for
some facts
and e x t e n s i o n s
of these
paper
[5] a number
results.
of
of
324
We take g r e a t p l e a s u r e Manin's
enumeration
the a c t i o n on divisors. numerical early
of
NS(X)
formulae
it i n d e p e n d e n t l y Igor D o l g a c h e v
of t r a n s l a t i o n
to our a t t e n t i o n references.
and Henry
discussions Pinkham
of Rome.
for their h o s p i t a l i t y .
w o r k and live.
for m a k i n g
of the
insight was h e l p e d
some years ago.
at an
We w i s h to thank
and for b r i n g i n g
for s u g g e s t i n g
This w o r k was b e g u n w h i l e
thank E n r i c o A r b a r e l l o
exceptional
this as a c o r o l l a r y
our
of
from Joe H a r r i s who had o b t a i n e d
w i t h Rick M i r a n d a
at the U n i v e r s i t y
institutions
law,
this r e s u l t
computation
by the s t a n d a r d
recovers
for the g r o u p
for h e l p f u l
a n u m b e r of debts.
is based on a g e o m e t r i c
A l t h o u g h our m e t h o d
stage by seeing
visitors
#
in a c k n o w l e d g i n g
Manin's
some u s e f u l
the a u t h o r s w e r e C.N.R.
We are g r a t e f u l
to b o t h
M o s t of all, we w o u l d
Rome
paper
such a d e l i g h t f u l
like to
p l a c e to
325
L I N E A R SYSTEMS
IN
~2
A classical
p r o b l e m in the theory of plane curves
is to study linear systems defined by base point conditions. Given a finite set
{pl,...,pk}
of points in the plane
p o s s i b l y infinitely near)
and a
with
one seeks the linear system
n >0
and
ni ~ 0 ,
curves of degree multiplicity
n
(k + i - t u p l e
(n;nl,...,nk)
in the plane passing through each
at least
ni
(some
In this paragraph,
L
of P. z
w h i c h is
p r i m a r i l y motivational,
we adopt the simplifying a s s u m p t i o n
the
i.e.
P. l
are distinct:
Nowadays we let at the points surface in
~2
of
L
X . and
Pi If
with
that
none is infinitely near.
~ :X _ + ~ 2
be the blow-up of the plane
and carry on the analysis on the rational H
is the p u l l b a c k of the class of a line
E i = ~ - l ( P i)
,
then pullback via ~ k with the linear system InH ~ i n i Eil on
gives a b i j e c t i o n X .
The classes
H, EI,...,E k
are a basis for NS(X), the Neron-Severi group k of X . If C ¢ In H - [ Eil , then following Nagata [6], we i=l refer to the (k+l) - t u p l e (n; n l , . . o , n k) as the numerical data or numerical
(2)
character of
C .
k K x = -3H + 7. E i i=l
we o b t a i n from R i e m a n n - R o c h
(3)
Since
k h 0(nH - 7 n i E i) i=l
that
[ = ~
k k (( n2 -7. ni 2) + (3n -C-[Ini i ) + 2) i=l
k + h l(nH -7. n i E i) . i=l
326
The
hI
term, c l a s s i c a l l y
the p o s t u l a t i o n
called the s u p e r a b u n d a n c e
that for generic choices of the
is a rather subtle invariant m e a s u r i n g of linear conditions
Pi
it is zero
,
the failure of the sets
imposed at each of the
The first example
to reflect
Pi
to
be independent.
every student sees is the linear system
of cubics passing simply through nine base points. this linear system is never empty.
We say
associated
conditions
if any of the equivalent
By
PI,...,P9
(L3), are
b e l o w holds.
9
i)
h0(3H - [ E i) = 2 i=l
ii)
h I (3H - [ E i) = 1 i=l
9
(4) iii)
The
{Pi }
are the base points of a pencil of plane
cubics. 9 iv) v)
If
9
E e 13H i ~ l E i I ,
There is a map
f:X
relatively minimal
(Recall that
X
then
3HIE ~i~l Pi
~IP 1 elliptic
making
( 2 )
F1
a
surface.
r e l a t i v e l y minimal means there are no exceptional
curves of the first kind lying in any fibre of the fibres
X
of
f
are the curves
are a n t i - c a n o n i c a l
: F =-K x
o
in
f .) In this case, 9 I3H - [ E i I so by i=l
327
Most on
X
when
k =9
r e s t of this paring
of this and
section
of s p e c i a l
ireducible
curve
that
PI,...P9
to m o t i v a t e
and g e n e r i c
cases
by
~
the
set of
this when
we a s s u m e 9 I3H - I Eil i=l
in
with
the g e o m e t r y
are a s s o c i a t e d .
arguments,
E
We d e n o t e
is c o n c e r n e d
we w i s h
the a s s o c i a t e d
a number
paper
In the
choice
by c o m -
k =9
that
10-tUples
there
To a v o i d is an
N=(n; n I .... ,n 9 )
satisfy
{ 5 )
i)
9 n 2 - ~ n 2 =-i i=l I
ii)
9 3n-[ n. = 1 i=l 1
iii)
We
denote
if
C£L N
terms
9 [ n i) = -2 i=l i=l 9 LN the l i n e a r s y s t e m In If- ~ n i E ilon X , and by i=l we call N =N(C) the n u m e r i c a l c h a r a c t e r of C . In (n 2
_
of a c u r v e
be r e e x p r e s s e d i)
ii)
(6)
Clearly,
CeL N ,
(3n-
the c o n d i t i o n s
(5) m a y
as
C2 = - 1
(-K X) C = 1
iii)
Condition
2
n i) -
Pa (C) =
( 6 .ii) any
C.C + K.C + 1 = 0 2
is e q u i v a l e n t
two of t h e s e
to
conditions
F.C = i imply
for a s s o c i a t e d
tSe third.
Pi
328
PROPOSITION
7
empty.
LN
i) C ii) iii)
If
If
Ne#
is s m o o t h
~(C)
and
a n d no o t h e r
in
"the v i r t u a l
so
That
The n o n - e m p t y n e s s
CI=C+D C = C!
arithmetic immediate
and
C'~C
with Th~s
genus from
curve
LN
C ,
is not then
.
ni
of o r d e r
in i) m e a n s ,
by s a y i n g
have
N
system
at
Pi
singularities-
is zero.
is i r r e d u c i b l e
L
points
superabundance
PROOF:
linear
rational
The c o n d i t i o n
that
the
an i r r e d u c i b l e
curve
has m u l t i p l e
REMARK.
then
contains
is the u n i q u e C
,
D gives
in
data
iii)
was
is i m m e d i a t e ,
then
effective
case,
that
the
classically
expressed
are e f f e c t i v e . "
since
and
from
( 3 ).
C C' = -i,
D~a
i). A s i n g u l a r
at l e a s t o n e w h i c h
ii).
in this
,
ii)
and
C
we m u s t
~ut t h e n
irreducible
yields
If
D = Q
curve iii)
has is
329
Now the
C. 3
if
Then
rji
S
S
suppose
reordering
S
the
r
C. , 3
9 3r.3 -i=l [
and
in a d d i t i o n
-> 0
for e a c h
for
{PI,...,P9} section
j ~i
are
of
X , so
so its n u m e r i c a l
S
associated. then
the m a p
is a section.
character
is in
Hence, LEMMA
8
i)
irreducible component
If
NE~
and C. 3
satisfies of
If in a d d i t i o n
and
CO
is a s e c t i o n In fact,
of a c u r v e
C0
CEL N (
then
5,
ii)
C =C 0 +D
and
where
E.C. = 0
CO
is
for e v e r y
D .
ii)
the c u r v e
and
{Pl,...,P of
one c a n
above
are a s s o c i a t e d ,
then
N(C0)e~
X . s h o w that
depends
CeL N . (cf.
}
[5])
only .
if the on
N
j
we have
is an i s o m o r p h i s m
is r a t i o n a l
implies
ji
r..3 l = 0
that
numerical
~ ---+ S.F~
In p a r t i c u l a r
9 E.Cj = 3rj ~"=
But
irreducible
by
(5.ii)
1
possibly
is any
~:~i+
) =l
£ [ m.C. where j=0 3 3 with numerical
C =
Rj = (rj;rjl .... ,rj9 )
9 i=l[ r01. = 1
Now
write curves
after
mnu = 3r0
C~L N ,
irreducible
£ 9 [ mj (rj ~ j=0
If
and
are d i s t i n c t
characters
Hence,
NE~
P. z and
are a s s o c i a t e d not on the c h o i c e
~ .
330
We conclude the p a t h o l o g i e s with
quartic and
P5
normal
these
P4
C . C
arise.
curve at
be chosen
in
~4.
PI,P2
Let
P5
P7,P8,P9
D . So
but there Moreover
(Again
P1
to
are
(1.4.i) section
and
Pl,...,Pq In the next
posed
and p a s s i n g shows
data
Let
checked
of
~ .
P4
intersection intersections C
and
D
and
is in
D+2L
satisfies (l.4.ii)
C
is an irreducible the cubic
irreducible:
a class of elliptic
systems
w i t h data
The d i f f i c u l t y rule out
though
of course
one suspects
is that
for n o n - a s s o c i a t e d
E.I = 0
for an infinite
examination
of this p r o b l e m
most
sets,
surfaces
in points
sets will have one m u s t
set of types
see Nagata
E
use Bezout).
We do not k n o w of any set of n o n - a s s o c i a t e d
w i t h this p r o p e r t y
can
the data
Of course,
(It is also
of linear
of
be the line
This data C+L
section, we will define
zero.
L
that
PI,...,P9
is not in
is always
through
Consider
two solutions:
is unique.
projection
be the residual
transform
quintic
that not all q u a r t i c s
C , (5;2,2,2,2,2,1,1,1,1)
the s u p e r a b u n d a n c e
For another
at
illustrates
be an i r r e d u c i b l e
are distinct).
for which
apparently
PI0
so the proper whose
D
be its residual
it is easily
PI0
a generic
are reducible.)
(at least)
the data of
through
it.
P3
P6
which
be an irreducible
Let
and
and let
(6; 2,2,2,3,3,2,1,1,1)
formal
C
- for example,
conditions
and
and
but not
Let
(An easy count of constants
through with
section w i t h an example
PI,...,P5
w i t h nodes
satisfying
of
which
five nodes
an elliptic
this
[ 6 ].
I
X
331
IRREDUCIBLE
Let surface
once one
ELLIPTIC
f:X ---+ C
with
structures
RATIONAL
be a r e l a t i v e l y
a section.
on the
fixes
SURFACES.
We w i s h
to d e s c r i b e
set of s e c t i o n s
a section
of
and u s e s
(Fl)n.s.
F1
of
space S
X
be the
over
SlE(Fl)n.s. on
operation
that
S10sS2
uniquely of
determines
X
becomes
group which
There
Let
in c a s e f~:Xn
S
the
becomes
S
a section
we d e n o t e
S
fixes
fibre
Formally
on the
fibre
homogeneous If
is a s e c t i o n a group
structure
Then define
denoted
$S
gives
1 curve
a
by r e q u i r i n g
an o r i g i n
be the o r i g i n a group
S
by
for
X
n
space
interpretation fibres
of
be the g e n e r i c
over
Thus
the set of for a
%(X)
of the
section,
isomorphism.
•
homogeneous
a bijection X~
(S2) I
S1 e s S 2
well-known
j-invariant
then
of the genus
of
S ,
on e a c h
[ 1 ])
+Sl
(SI) I +Sl
---+ ~ = S p e c ( k ( C ) )
If we c h o o s e
of
a principal
is a s e c o n d
Specialization S
(cf. Since
we d e n o t e
sections
speaking
a principal
as the o r i g i n
operation
on the
.
of g r o u p
fibre"
points
group.
S 1 =S.FI
S1
Roughly
by
is n a t u r a l l y
(SIS S S2) ~ =
commutative
"fibre
algebraic let
a class
solve
( 9 )
sections
X
whose
group
group
This
so t a k i n g
(Fl)n.s.
This
IcC
of
.
set of n o n - s i n g u l a r
for a c o m m u t a t i v e
is a s e c t i o n
X
elliptic
it as the o r i g i n
t h e n one can add any two s e c t i o n s let
minimal
n
is n o n - c o n s t a n t .
f i b r e of
between
k(C)
as above,
and
let the
then
details,
this
f
-valued
and s e c t i o n s
(k(C))
For m o r e
X
of t h i s
S
of
points X
.
specialization bijection
see M a n i n
[3 ] .
332 For the r e m a i n d e r
of this paper,
relatively minimal
rational
X
by r e s o l v i n g
can be realized
curves
in
p2
whose
base points will, X
in general,
say that
equivalent
X
i)
All
be a
the base locus of a p e n c i l is smooth.
be i n f i n i t e l y
Then GI
of c u b i c
(Some of the
near.
proof a good r e f e r e n c e
is i r r e d u c i b l e
conditions
f:X + ~ i
surface w i t h a section.
generic m e m b e r
w h i c h we quote w i t h o u t
We will
elliptic
let
For facts about is M i r a n d a
[ 4 ]).
if e i t h e r of the f o l l o w i n g
hold.
fibres
FI
of
X
are irreducible.
(i0) ii)
The base points of the pencil curves three
To see that these
giving
are e q u i v a l e n t ,
n e a r base p o i n t
the p r o p e r
transform
assume
distinct. G
=Q.L
theorm
Let
so in fact if three some
#L n GA = 3 Pi
member with
and no
and
L
lie on a line
theorem
L G
Q
then
implies
Q.L
Hence we may
of the pencil
of the pencil linear.
are
and suppose
Then B e z o u t ' s But
#(G u n Gl) = 9
are collinear.
is the q u a d r i c Q.L
is an
fibre c o n t a i n s
#(L nG~) s 3.
and ,
that there
divisor.
and three base points
five of the r e m a i n i n g
•
6
if some
PI,...,P9
be a smooth e l e m e n t
#(Q n Gl) s
and a g a i n B e z o u t ' s {S l }
of an e x c e p t i o n a l
is a r e d u c i b l e implies
are d i s t i n c t
first o b s e r v e
if and o n l y
the base points GI
X
of cubic
lie on a line.
infinitely
as well
rise to
GI
contains
is a m e m b e r
Conversely,
through 8 of the
Pi
of the pencil
333
We remark the r e d u c i b l e cubics
such
izations and
that i r r e d u c i b l e
cubics X
have c o d i m e n s i o n
are generic.
of the i r r e d u c i b l e
(26)
.
X
L~MMA Ii
section
rational
elliptic
If
X
by,
is i r r e d u c i b l e
and
then the f o l l o w i n g C
is a s e c t i o n of
ii)
C
is an e x c e p t i o n a l
iii)
The n u m e r i c a l
A section
(6.iii)
and an e x c e p t i o n a l
in
imply iii).
s e c t i o n of
X
o n l y if
C 2=-I
PROOF:
Write
each
Ci
and
.
of i r r e d u c i b i l i t y
considered
in
is an e f f e c t i v e
N(C) curve
is in
curve
~ .
satisfying
c u r v e of the first kind (6.iii).
C.F = I), Moreover
(6.ii)
is an
and
irreducible
Since two of the c o n d i t i o n s
it is i m m e d i a t e
is i r r e d u c i b l e
(i.e.
that
iii)
i)
ii) and that b o t h
i) follows
from
and
C
is an e f f e c t i v e
then
C
is i r r e d u c i b l e
in this c a s e
C
numerical if and
is a section.
k C = [ C i + [ F. where each Fj is a fibre and i=0 j=l ] is an i r r e d u c i b l e curve not equal to a fibre. Since X
is i r r e d u c i b l e , i =i
character
(6.i) and
X
(14)
c u r v e of the first k i n d on
The critical i m p l i c a t i o n If
C
in
X
is an i r r e d u c i b l e
(6) imply the third
L E M M A 12.
surfaces
are e q u i v a l e n t :
i)
satisfying
systems
is e x p r e s s e d
PROOF:
curve
2 in the space of all
The i n t e r e s t of the h y p o t h e s i s
the p r e c e d i n g
X
In fact since
We shall give some o t h e r c h a r a c t e r -
from the point of v i e w of the linear
on
exist.
Ci-F > 0
C0-F = 1
for each
Since
CO
i .
Hence
is i r r e d u c i b l e ,
C •F =i
implies
it is a s e c t i o n
334
hence and o n l y of
C
if
C0 2 = - i
£ =0
;
.
this
gives
in t u r n
13.
If
a bijection
X
N=N(C)
so
C 2 =-i
to the
if
irreducibility
between
#(X)
when
section
X
and
shall often
is i r r e d u c i b l e
by g i v i n g
the map
another
C
~ N(C)
# . s p e a k of t h e c u r v e
and
Ce~(X)
We c o n c l u d e
characterization
of i r r e d u c i b l e
.
LEMMA
-l~
X This
PROOF:
and Shioda (15)
where X
C 2 =2£ -i
is e q u i v a l e n t
is i r r e d u c i b l e
In v i e w of t h i s w e
X
then
.
COF~DLLARY
this
But
is i r r e d u c i b l e is a c o n s e q u e n c e
[ 8]
for e l l i p t i c
rankz(NS(X))
mI
if and o n l y
of a g e n e r a l
surfaces
rank
~ (X) = 8
f o r m u l a of
f:X ---+ C
..
Tate
[ 9]
.
= r a n k Z ~(X) + 2 + I (m x -l) lec
is the n u m b e r
is r a t i o n a l ,
if
of c o m p o n e n t s
of c o u r s e ,
rank(NS(X))
in the =i0
fibre
Fl
.
When
335
THE
CROUP
L A W ON
Our (X)
~ •
first goal
is to w r i t e
in t e r m s of the a d d i t i o n
LEMMA
16
PROOF: linear
SlesS 2 =
Denote
by
the
of d i v i s o r s
~ l ( ( S l ) l - S l) +
SlesS 2
equivalent
is a s e c t i o n
calculation
yields
induced
and a s e c t i o n not o n
X
.
t h a t on
for s o m e
by
of t h i s
lemma
isomorphic.
section
•
#
or o n
~(X)
in b o t h c a s e s .
"universal"
group
l a w on
X
,
the g r o u p
precisely,
~ .
surface of
X
S ,
structures
H e n c e we m a y
and t h e n We d e n o t e
More
the g r o u p
character
the e x c e p t i o n a l
(0; - 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) on
Since , and a s h o r t
of an i r r e d u c i b l e
are
structure
e .
is that
fixed
character
(9)
e .
for
for all as the o r i g i n
of
X , SlesS 2
Ii
Lemma
hand S
~l
or e q u i v a l e n t l y
this m e a n s
by the c h o i c e
to d i f f e r e n t
and by
the c o n d i t i o n
On the o t h e r
with
down the
Then
Fl
+i
that
= -i
of
on
on the n u m e r i c a l
and
by
means
2
on
s = (S 1 + S 2 ) S - S l . S 2
only
once
operation
.
S 1 +S 2 -S +eF
corollary #
where
Fl
But
~S
structures
NS (X)
depends
corresponding
group
to
.
the v a l u e
on
S
in
((S2) ~ -Sl)
, (SlesS 2)
An immediate structure
on
(S2) I
(SlesS2) l ~ l ( S l ) l + ( S 2 ) I - S l is l i n e a r l y
the g r o u p
s u m of d i v i s o r s
(SlesS2) I = ( S l ) l + S
(SlesS2) I -Sl
+
(Sl+S 2 - S + e F) +
equivalence
down
fix
divisor
E1
s p e a k of the
the c o r r e s p o n d i n g we c a n n o w w r i t e
336
THEOREM =
17 i)
( n ; n l , . . . , n 9) •
(m;m I ..... m 9)
(n+m+3~, n l + m l +
l - ~, n 2 + m 2 -~, 9 ~ = n I + m _ - n m + ~ n. m. + 1 I i=l ~ 1 1
where
ii)
O
( n ; n l , . . . , n 9) =
PROOF: and the
i)
Simply
set
S = (0;1,0,...,0) formula
ii)
Let
Hence
of the
implies
in L e m m a for
with
that
; n l , 2 + 2 n l - n 2 .... , 2 + 2 n l - n 9)
S 1 = (n;nl,...,n9)
lemma
S2 = O S1
i)
(6 + 6 n I - n
n 3 + m 3 - e , . . . , n 9 + m 9 -e)
, S 2 = (m;ml,...,m9)
16, c o m p u t e
e
and e v a l u a t e
S 1 OsS2
the n o t a t i o n
(m;ml,...,m9)
above.
Then
is the u n i q u e
SlO S 2 = solution
E1 in
of
n +m +3e =0 nl+ml+l+e =-i n.+m.+e =0 l 1
But the r i g h t
hand
such a solution COROLLARY
18
sid~ of the
using i)
9roup
ii)
of
commutes EXAMPLE
with
( 5 ) for
The m a p
is a s u r j e c t i v e The a c t i o n
, for
S8
~
i = 2 ..... 9
formula
is e a s i l y
checked
to be
( n ; n l , . . . , n 9) ~ Z/3Z
by
( n ; n l , . . . , n q)
---+ n
homomorphism. on
~
by p e r m u a t i o n s
of
(n I .... ,n 9)
• .
"19 e (0;0,-i,0,0,0,0,0,0,0) 8 (i;0,i,i,0,0,0,0,0,0)
= =
(6;0,3,2,2.2,2,2,2,2) (5;0,i,i,2,2,2,2,2,2)
8
(2;0,1,1,1,1,1,0,0,0)
=
(4";0,1,1,1,1,1,2,2,2)
8
(3;0,2,1,1,1,1,1,1,0)
=
(3;0,0,1,1,1,1,1,2)
(mod 3)
337
The o r b i t s
of t h e s e
yield
all
[ 2
, Proposition
240 e l e m e n t s
With of
solutions of
#
under
the a c t i o n
with
n I =0
S8
. (cf. M a n i n
26.1])
a little
more
effort
we can
list
the e l e m e n t s
# .
PROPOSITION
20
Fix an 8 - t u p l e
(Manin[
3 , Theorem
of i n t e g e r s
A =
5])
( a 2 , . . . , a 9)
and
set
9 [ a. 9=2 3
NA=
9 M A = [ (aj) 2 + [ aja k - N A j=2 2_<j
Then
of
-
(3 M A) ; M A + N A -i, M A - a 2 , . . . , M A -a 9)
if we let
Ck = C eC @...@ ~ k times
C , we h a v e i
a2 a9 E A = E 2 • ... • E 9
PROOF:
The
exceptional
formula
is e a s i l y
divisors
E2
By i n d u c t i o n
and
symmetry
of the P r o p o s i t i o n
for
EA
checked
to
that
then
in the n o t a t i o n
of T h e o r e m
(3(MA+(NA+n2));
EA A' =
to c h e c k
is e q u i v a l e n t
or e q u i v a l e n t l y
so t h a t
Let
it s u f f i c e s
E A'
+ 1 = NA+ n2,
E9
when
EA • E 2
(MA + N A + n 2 ) +
is g i v e n
that
the
If we add ~ =mA+NA-I
assertion for
EA
and
E2 ,
+ 0 - 0 - ( M A - n 2)
by
(NA + i - i ) ,
(MA + N A + n 2) - n 3 . . . . .
(a 2 + l , a 3 , a 4 , . . . , a 9)
to the a s s e r t i o n
E A @ E 2 = E A' 17
is one of the
(MA+NA+n2)
-
(MA + N A + n 2) - n 9)
(n2 + i ) ,
338
But
NA,
= N A +i This
of
the
and
~ = Z 8
divisors
E2, . . . ,E 9
COROLLARY
21
ii)
~/~
PROOF: the
i) If if
Part
(17.i).
of
~
us
hence
a numerical
generated
by
N (C) = ( m ; m l , . . . , m 9) E ~ ,
m--0(mod
ii)
is
in i)
E A ~ E 2 = E A' description the
then
exceptional
C ~
3)
C =E A
,
m'
we must
(22) fix
SinceN(C~
immediate
then
hand,
from
suppose
we must
i)
and
the
( 18.
addition
m =3m'
i))
and
formula
If w e
are
have
= MA
m I = MA+
N A -I
m i =MA-a
i , i~2
have,
a i =m'
-m i
, i~2
these
ai
and
,
from
is i m m e d i a t e
On the other
to have
Let us
gives
,
= Z /3Z
necessity
Hence
= MA+NA+n2
Proposition
subgroup
if a n d o n l y
MA,
we have
by
prove ( 5 )
9 9(m') 2 = [ m 2 -i i=l 1 9 9m' = [ m. + i i=l 1
that
m' = M A
and
m I =M+N-i
339
From the second we find 9 m I = 9m' - [ m -i j=2 ] 9 =
m'
-
[
(m'
9 =2 9 =m'+[ j=2 =
m'
-
-i
mj ) a. 3
+N-I
Hence it will suffice to show that
m' = M
.
Now rewrite the
first equality as 9 9(m') 2 = ml 2 + [ (mj) 2 - i j=2 9
=
(m'
+N-i)
2 + [ (m' - a j ) 2 9=2
-i
After e x p a n d i n g the right hand side and s i m p l i f y i n g expression,
the resulting
we o b t a i n
9 2 2m' = N 2 - 2 N + [ a. 9= 2 3 9
9
: ([ a~) 2 + [ j=2 J
=
aj 2
-
2N
j:2
2M
as required The simplest section w h i c h has degree congruent to zero modulo three and w h i c h (n2,...,n 9)
is
G =
n
not
is symmetric
(4;3,1,1,1,1,1,1,1,1).
Using
in
(20.i)
~0 we
find
G3 =
G 2=
(84;
35,
(23;
15,
i0,I0,i0,i0,i0,I0,i0,i0)
27,
27,
27,
(The last
equality
COROLLARY
23 Let
unique i)
8-tuple
if
of i n t e g e r s
off
27,
from
3),
then
C =E A
MA +
n ~ 1 (mod 3),
27)
E9 .
~
Then
there
such
that
is a
and
N A - i,
then
= E 2 eE 3 ~...e
(22)
A = (a2,...,a 9)
(n; n l . . . , n 9) = (3M A + 4 + 9 N A ; MA-a
27,
N ( C ) = (n; n l , . . . , n 9 ) c
n H0(mod
if
27,
can be read
(n ; n I ..... n 9) = (3MA; ii)
27,
and
MA-a2,
C =G eE A
....
MA-a9).
and
M A + N A + 3 - 3 N A, M A - a 2 + I - 3 N A, ....
9 + 1 - 3 N A)
iii)
if
n H2(mod
3),
then
( n ; n l , . . . , n 9) = ( 3 M A + 3 2
C =G 2 eE A
+I8NA;
and
MA+N A +15-6N A
, MA-a
2 +10-6N A
,
.... M A - a 9 + I 0 - 6 N A) PROOF: those
The about
addition
assertions the
about
f o r m of
formula
representatives
The p r e c e d i n g In [ 5 ] ,
results
above
( n ; n l , . . . , n 9)
of of
enumeration
we o b t a i n
is i n d e p e n d e n t
of are
f o r m of
C from
follow (20)
from
( 20.i);
and the
(17.i)
A description coset
the
differing #/~
is g i v e n
paramet
its d i o p h a n t i n e
rization
of
[ 3 , Theorem
of l i n e a r i z i n g
description
consequences.
on the c h o i c e
by M a n i n
has the e f f e c t
a linear
formal
only
of
#
~ . which
and of w h i c h
the
6].
341 RELATIONS W I T H
AUT0(X)
Fix a rational ell~ptic necessarily a section
irreducible) S
of
X
this induces on
surface
with n o n - c o n s t a n t
and denote by
~(X)
and by
Aut0(X) of
Cl
X
fo~ = f .
be the point
F~ ,
we set
continuity
j-invariant
+l
C.Fl
.
If
C
If
this
points of the fibre
Fl
i(C) (Q) = Q + ~ C~
by
C
X ,
is a n o n - s i n g u l a r .
to an a u t o m o r p h i s m of
just t r a n s l a t i o n
automorphisms
is a section of
Q
Then
X .
i(C)
extends by
Informally,
fibre-by-fibre.
o
denote the f i b r e - p r e s e r v i n g
taking fibrewise
inverses:
if
Q
let
point of
Since
i(C)
i(C)
fibre p r e s e r v i n g we o b t a i n an inclusion of groups Let
and
the group o p e r a t i o n
be the set of f i b r e - p r e s e r v i n g
i.e.
(not
8 and O the group o p e r a t i o n s
induces on the set of n o n - s i n g u l a r Let
f:X -~ ~ i
invDlution on
is clearly
i:#(X) X
is n o n - s i n g u l a r
is
---+ Aut0(X).
induced by on
Fl , o(Q)=-iQ
Then, PROPOSITION
24
Aut
is @enerated by PROOF:
Given any
section of Aut0(X) of
X
o
X
fixing
(X) & i(~(X)) e Z/2 0 and the action of ~ on
ae A u t 0 ( X ) ,
and let S ,
fixing the origin.
constant,
i(~(x))
let C = a(S)
8 =i(@ C)a hence
where the
.
induces
Then
factor
oi(C)o-l= i(OC).
w h i c h is again a 8
is an element of
automorphisms
Since we assume
this means that for generic
is by
Z/2
of the fibres
jl =j(Fl)
l, 81Fl
is not
is given by the
342 identity
or by
or
So
~ .
coi(C)c(Q)
~ =i
Hence
(C)
that
result
if
M
that
Hill
the
X
natural when
then
determines
the r e p r e s e n t a t i o n
of
Q~F~
,
map
on
Aut(X)).
(C')
= C • C'
surface
and t h a t
The n e x t
Aut0(X)
to
on all of
M(i(C))
§3
as in
if
restricted
is an i r r e d u c i b l e
S=EI,
identity
= i(S C) (Q)
not be f a i t h f u l
NS(X),
section
M
that
the
Moreover
is i n j e c t i v e
our o r i g i n
by
is e i t h e r
~ =i(C)o~
implies
C'E#(X)~
let us a s s u m e
itself
-I Q) = ( S C) I +l Q
(The m a p
Observe
8
or
---+ A u t ( N S ( X ) )
Aut0(X)
Now
;
= -I(Cl This
M:Aut(X)
-I
proposition
Aut(NS(X))
given
.
PROPOSITION
25
Fix the o r d e r e d
basis
(H; - E I , . . . , - E 9)
of
NS (X) i)
If
matrix
C =
(n; n l , . . . , n 9 ) e
coefficients
of
CHH = i0 + 9 n I
#
C
and
C =M(i(C))
then
the
are
CHE 1 = 6n I + 6 - n
CHE" = 3 ( n I + h i + l ) 1 i>2
CEI H = n
CEIE 1 = nl
CE. H = 3(nl-nj-l) 3
+n
CEjEI
j a2
and
for
i >-2
CEIE 1 = n i ,
= n I +l+(nl-nj+l)
ja2
and
j ->2
ia2
-n
343
n i + (nl-n ~ +I),
CEj E i =
I
ni
ii)
The matrix
i ~j
J
,
coefficients
i =j
of
~ =M(o)
l
=17
l
=0
l
=6,
i~2
l
=0
l
=-i
l
-0,
i>2
HH
EIH
H.EI
EIE 1
[E.H = 6 3
formula
M(i(#))
EI,...,E 9
on
does the same for
H.Ei
EIE i
[E3E 1 • =0
The addition
(17.i)
[E3E l = ~2, i~j ia2 , j~2 "" [3, i=j
gives the action of
and the inversion
M(~)
. Since elements
fix the class of a fibre r this reduces a straightforward REMARK:
Ei
changed-of-Sasis
l) The matrices
were obtained
Proposition
by
Aut0(X )
to the exceptional
to
divisions
who based his proof of
Rick Miranda
and Joe Harris
these matrices
of all sections
are obtained
of
( 17 .ii)
the p~oposition
the matrices of the
the matrices
the matrices
[3 ]
later rediscovered
By diagonalizing
to compute
Manin
formula
computation.
associated
20 on them.
(unpublished) [ .
are
Ei's Ce~
by direct geometric
and the matrix they were able In both cases,
arguments.
344
2) continue of
If we do not suppose
to assume
C =M(i(C))
C~(X)
§7
that
OF T O R S I O N
We c o n t i n u e
by
section, ~
is a section,
see [5]
section
where
of
by this
(not n e c e s s a r i l y
B
where
can c o a l e s c e at
elements (C, 8 C)
B = on on
. X
diagram
only
f:X--+ IP 1
and to d e n o t e on
Then
and the b r a n c h
X
.
of
R
disjoint
distinct if
R
Let
of
locus on
from
is s i n g u l a r
a rational
morphism
R and
S . Q1
on
and
FX0
.)
2-torsion
b e t w e e n pairs of s e c t i o n s
everywhere
to e a c h pair t h e i r c o m m o n the g e o m e t r y
R
t o r s i o n points
QX0
R
is a ruled
is a s e c t i o n of
is a b i j e c t i o n
and s e c t i o n s
(For m o r e d e t a i l s Define
X
surface
is just the locus of p r i m i t i v e
There
g i v e n by a s s o c i a t i n g R .
10
-i(~)
X
on
involution
S = y (S)
is a smooth t r i s e c t i o n
The curve
elliptic
involution.
minimal)
(Observe t h a t two g e n e r i c a l l y Q'I
for any
R
X
B + S
S
minus-one
x
has the form
then the m a t r i x
.
to fix a rational
and an o r i g i n
be the q u o t i e n t
but
POINTS
the c o r r e s p o i n d i n g
surface
irreducible,
is still g i v e n by the p r o p o s i t i o n
For more details
CURVES
with
E1
X
of d o u b l e P
which
tangent image
covers
to
B
C =y(C) see
fits in a
[
on 7 ])
345
by blowing disjoint
down all e x c e p t i o n a l
from
S
neighbourhood model
of
of
LEMMA 26" X
of
R .
a section
Then S
and
Since R'
r
divisors
R'
which
and
are
in a
is the r e l a t i v e l y
of s e l f - i n t e r s e c t i o n r
R
is an i s o m o r p h i s m
~2 = -2 = 2(S 2)
The m a p
on
minimal
r(S)
is again
-2, R' =IF 2
is an i s o m o r p h i s m
if and only
if
is irreducible.
PROOF:
Clear
from the construction.
As g e n e r a t o r s class of a generic T 2 =2,
T.G = I
the unique
r(S) e JT -2GJ disjoint
in general between
r(B)
point on
Define
Tn
that an
r(B)
is and
and is
is no longer
is a d i c t i o n a r y
and the K o d a i r a
We will 12
X
content
fibre p r o d u c e s
be i r r e d u c i b l e
T'n
hence
could
and let
n-torsion
is the curve
T' -E 1 = 0 n
above.)
r(B)
r(S)
types of
ourselves
here
an o r d i n a r y
r(B)
where
in its fibre, remarked
X
to be the locus of
T n =Tn' + E l and
of
(so
a trisection
While there
the
self i n t e r s e c t i o n
is still
trisection,
G
r(S) 2 = - 2 ,
of n e g a t i v e
the s i n g u l a r i t i e s
N o w let
points
Since
~2
fibres of
we take as usual
and of a fibre
, r(B) c J3TJ
a smooth
the example
double
of
r(S)
the d e g e n e r a t e with
T
G 2 =0)
Since
from
N S ( ~ 2) ,
section
and
section
of
(A point of
S =E 1
points
of
of n o n - t r i v i a l T
n
nE 1
would
not lie on the section
as usual X .
Then
n-torsion be singular
E1 ,
as was
346 LEMMA
ii)
27
If
PROOF:
i)
C i)
n-torsion and if
9 Tn(X) elin2 - I ) { 3 H -[
j=2
is any section disjoint
from
Clearly
,
TA(X)-F = n 2 -i
points on an elliptic
i >i
Immediate
E 1 , TA(X).C = (n 2 -i)
the number of non-trivial
curve.
As above,
T~(X)-E i = E 1 E~ = n 2 -i '
ii)
Ei; I
by
T~(X)-E 1 = 0
(Z0)
1
from I) and
"
C-F = 1 .
We remark that part i) of the Lemma remains valid if we suppose ~:X---+~ 2
X
is any rational
is a blow-down,
(some possibly cubics and
COROLLARY
that
infinitely
PI,...,P9 near)
surface,
are the base points
of the associated
pencil of
E i = ~ - I ( P i)
28
Let
Tn(X )
be the locus of primitive
'--'
points of
elliptic
X .
Then
n-torsion
9"
T*(X)E 17(n)n
(3H -[j=2 Ei) I
where
y(n) = 2n2dl 'nd2
~7 REFERENCES
i.
Kodaira,
K.: On C o m p a c t
Math.
(1963),
77
Analytic
563-626;
ibid 78(1963),
2.
Manin,
Yu.
I.
: Cubic
3.
Manin,
Yu.
I.
: The Tate Height
Its V a r i a n t s 59 4.
(1966),
Miranda,
6.
Nagata, Kyoto,
Amsterdam,
II,III,
of
1-40. North-Holland(1974).
of Points A.M.S.
Ann.
on an A b e l i a n
Translations,
Variety.
Set.
2,59
82-110.
R.
Morrison, Surfaces
Forms.,
and Applications.
: On the S t a b i l i t y
w i t h Section 5.
Surfaces,
~ Thesis
(1979)
I. and Persson,
U.
of Rational
Elleptic
Surfaces
M.I.T. : Numerical
Sections
on Elliptic
(to appear). M.
: On Rational
Ser. A. Math.
32
Surfaces, (1960),
I, II. Mem.
351-370;
ibid.
Coll.
Sci.
Univ.
33(1960-1961),
271-293. 7.
8.
Persson,
: Double
Coverings
168-195
in : A l @ e b r a i c
Shioda,
T.
24 9.
U.
(1972),
Tate,
J.
Geometry,
: On Elliptic
Modular
Springer
of General
L.N.M.
Surfaces,
687
J. Math.
Type, (1977).
Soc°
Japan,
20-59.
: On the C o n j e c t u r e
and a G e o m e t r i c 1-26.
and Surfaces
Analogue,
of Birch and S w t n n e r t o n - D y e r
Sem.
Bourbaki,
Exp.
306,
(Feb.
1966),
On the Kodaira D i m e n s i o n of the Siegel M o d u l a r V a r i e t y
by David M u m f o r d
Let
~g r e p r e s e n t the q u o t i e n t of Siegel's upper h a l f - s p a c e
of rank g by the full integral symplectic group Sp(2g, ~) : this is known as Siegel's m o d u l a r variety,
or as the moduli
of g - d i m e n s i o n a l p r i n c i p a l l y p o l a r i z e d abelian v a r i e t i e s p.p.a.v, (i.e., [FI]
below).
space
(called
A g has been shown to be a v a r i e t y of general type
Kodaira d i m e n s i o n = dimension)
for various g's:
Freitag
p r o v e d this first if 241g; Tai ~] p r o v e d this recently for
all g ->- 9.
On the other hand,
g ~ 5: D o , ~ % g ~ 3.
Ag
is known to be u n i r a t i o n a l
[D] for g = 5, Clemens [C]
for
for g = 4, c l a s s i c a l for
The purpose of this paper is to refine Tai's result,
showing:
Theorem:
Ag
is of general type if
g > 7.
Note that this leaves only the Kodaira d i m e n s i o n of to be determined.
A6
still
We shall use results of Freitag and Tai in a
crucial way, but the idea of the proof is a direct a d a p t i o n of the proof type if
[H-M] by Harris and the author that
g ~ 25, g odd.
M
In that proof the divisor
which are k~fold covers of ~ i
,
k = ~g+l
is of general
g Dk
of curves
, is shown to be linearly
e q u i v a l e n t to nK-(ample divisor)-(effective
divisor).
Here we prove the same thing except that the role of D k is taken by the components of
No,
where
~9
These sets
~
were i n t r o d u c e d by A n d r e o t t i and Mayer
studied r e c e n t l y by B e a u v i l l e
~] .
[A-~ , and
I w a n t to thank Beauville
very much for s t i m u l a t i n g d i s c u s s i o n s w h i c h led me to this result.
At the same time,
I w o u l d like to raise the q u e s t i o n
which seems very i n t e r e s t i n g to me: p o l y n o m i a l in theta constants, from theta series coefficients)
or other m o d u l a r forms c o n s t r u c t e d
(with q u a d r a t i c
whose zeroes give
is there an e x p l i c i t
forms and p l u r i - h a r m o n i c N O with suitable m u l t i p l i c i t i e s ?
A l t h o u g h i m p o r t a n t steps are taken in this d i r e c t i o n in Andre0tti-Mayer
[A-M] and B e a u v i l l e
because the "theta nulls"
C(r,~,z)
[B], this is not a n s w e r e d are not in general m o d u l a r
forms - - they are theta series whose c o e f f i c i e n t s are not pluri-harmonic;
esp. you cannot form a m o d u l a r
~2~/~u~'s alone w i t h o u t using mixed d e r i v a t i v e s Finally,
form out of the ~2~/~Uk~U ~
I w a n t to m e n t i o n the related results of S t i l l m a n
(based on earlier ideas of Freitag carries h o l o m o r p h i c
~2])
[S]
w h i c h prove A ~ g (4g-6)-forms for g ~ /~. These results are
directly based on the use of theta series.
too
350
§i.
A partial
compactification
of the Siegel m o d u l a r
Satake 's c o m p a c t i f i c a t i o n set-theoretically,
in the u n i o n of Ag•
The K o d a i r a
Ag*
dimension
=
Ag ~
of
A
Tai has
that a p l u r i - c a n o n i c a l
above
to study
A
IA
g
the full
Namikawa define pair
a rank
abelian
Variety
constructed
and D the
However, "no
so we do not have
this precise of
form with
in a minute.
Ag ~ A g _ l
by the author
~]
g e o me t r i c a l l y ,
first
and by let us
as follows:
g-dimensional
it is a
variety
and D
is to be the limit of a g - d i m e n s i o n a l
limit of its
theta
divisor).
G
is
as follows:
i) let B g-I be a theta
(g-l)-dimensional
p.p.a.v.,
E c B its
divisor
2) let G be an a l g e b r a i c 0 3)
regular
of a p.p.a.v,
G
A~. g
differential
space
is a c o m p l e t e (i.e.,
of
g
is a b l o w - u p
this
1 degeneration
divisor
~
We will make
[I] and studied
G
on p l u r i - c a n o n i c a l
is e v e r y w h e r e
To d e s c r i b e
(G,D) w h e r e
is an ample
'
to work with
by Igusa ~].
"
Ag.
The space we w a n t introduced
g-i
consists,
(g+l)-strata:
is based
g
on a d e s i n g u l a r i z a t i o n
poles
Ag
Ag_l ~ .... ~. A 0 .
differentials shown
of
variety.
>~
Considering
associated
>G
m G as
a
group w h i c h >B
II~ - b u n d l e m
Ipl-bundle:
is an e x t e n s i o n
>0. over
B,
let
'G b e
the
of B by •
m
:
351
G
c
~
B
Then
~-G
equals
G0~t~
, the union of 2 sections
of ~ over B.
4)
Then G
is to be the non-normal
glueing
5)
Note
%, <
variety
with a translation
obtained
by a point
by
b £ B.
that on %-<
~ n-l(E)'
E algebraically
equivalent
to 0 on
~-I(E-H b ), for a unique b I £B1 Thus ~0+w-l(Eb,)
Let ~ = sequence
for
Via the Leray spectral
~, we see that ~ +~-I(E)
I~I~ 0 ~
is chosen
+ ~-I(E ) .
@~(~+~-I(E)).
~0+~-l(Eb,), Then
- ~
@B(E)
and
h0(~)
span the linear ~I~
~ @B(Ebl),
to be b I (and only then)
can be descended
= 2 and that system so if
L on G.
Choose
such an L and let divisor
in
ILl.
We now define (i. i)
--(i)
~g
=
b
the line bundle
to a line bundle
D = the unique
l~I.
coarse moduli space of p.p.a.v.(A,0) of dimension g and their rank 1 degenerations
B
352
As first shown ~y ~gusa,
this space exists,
variety,
and is essentially
AgikAg_l
in
A
g
A*g
the blow-up
along its boundary
and a divisor
A parametrizing
is a quasi-projective
of the open set
Ag_l.
the rank
~(1)g is the union of 1 degenerations.
Via
the map (G,D) 4
> (B,E)
is seen to be fibred:
the divisor
& I
(i. 2 )
~t
fibres B/Aut (B,E)
Y
Ag-1 Analytically,
we may consider
degenerations
of the abelian
~(i) g
to represent
variety
AQ(t)
precisely
the
with period matrix
Q(t) when:
Im ~II and ~ij' Then
B = B
i >i
where
or ~(i)
>=
}
as
t
>0
j >i, have finite limits is the lower right block of the limit
n(1) '
~(o)
:
(
ioo
~o
to9
~(I)
and b is the image of the vector B (i)" n
~ = (~12(0) ~13(0),-'',~ig(0))
To find D, we must translate
O~(t)c A~(t)
as
t
in , 0.
353
Thus @~(t) = {
Translate
0n(t)
zer°es °f
@(z'~) =
b(t) ,
the image of
by
n6[~} e~itn~(t)n+2Zi%'z ' g
(~1~(t) ,0,-..,0) :
{~i ~ n.n .~. •(t)+2~itnz}} Tb(t) (0~(t)) ={zeroes of ~ e zi(n~-nl)~11(t).e i,j~l,l i 3 13
Then eZi(n12-nl)~ (t)
~0
unless n I = 0 or I, hence the limit is {~i [ ninj~ij(0)+2~i [ n.z.}/ e l, j>z j<2 3 3~II+e2~izl-e2~ij<--2 [ n.~.3 13~ "~
zeroes of
n2 ,...,ng£ 7z (I.13) ={zeroes where
of
~(z(1),~ (I)) + e2~iz' .O(z(1)+~, ~(i))}
z (I) = (z2,...,Zg)
Interpreting
e 2gizl
as
the
is the analytic coordinate on B (1)" algebraic
coordinate
in
the
fibre
of G, and E as the zeroes of ~(z(1),~(1)), this is m immediately seen to be D if L is suitably defined.
Next, let
~(i),0 be the open set in ~(i) parametrizing g g those pairs (A,@) or (G,D) whose automorphism group is the minimal one,
{~i}. More precisely,
automorphism of A (or G) mapping the form
x l
>-x+a, some a*.
the only non-trivial 0
Then
(resp. D) to itself is of ~(i),0 is locally isomorphic g
We have not normalized 0 and D to be symmetric. On the other hand, we have not fixed an origin either, so the pairs (A,e) and (A,0 c) are isomorphic by translation by c, and define the same point of ~(i). g
354
to the u n i v e r s a l d e f o r m a t i o n space of a smooth of d i m e n s i o n g(g+l)/2. subset of
Ag
(or (~,D)), hence is
Analytically,
of points which are images of
stabilizer in Sp(2g,~) d e s c r i p t i o n of subset of
(A,@)
T(1) g
~i)'" g
are just
(+I).
A0 g ~ £~g
Likewise,
in Ash et aI[A-B-K-~
is the open whose
using the analytic
K(I) ,0
is the open
g
of points w h i c h are images of points in
~g/U~){6e } whose stabilizer in the n o r m a l i z e r of the first b o u n d a r y c o m p o n e n t is just in particular,
(+I).
(Compare Tai
[T],
those i~ c o n s t r u c t e d from a
b 6 B not of order 2.
§
).
This set includes,
(B,H) 6 A 0 and a point g-i
We are now in a p o s i t i o n to state one of
the m a i n results of Tai's paper
[T], in the form in w h i c h we need
it:
T h e o r e m 1.4
(Tai).
a)
codim
b)
F(%,O(nK))
I f g > 5, then
(A(1)-i~(1),0) > 2 g g --
an__dd =
F'A(1)'0,O(nK)) t g , _if _
n _> I.
This means that a p l u r i - c a n o n i c a l d i f f e r e n t i a l with no poles on ~(i),0 g of
is e v e r y w h e r e regular on a full d e s i n g u l a r i z a t i o n
g
A*. g The second result we need is the c a l c u l a t i o n of
follows from the theory of Matsushima,
Pic~U)
Borel, W a l l a c h and others on the
low c o h o m o l o g y groups of d i s c r e t e subgroups of Lie groups. cular,
the results of Borel
F c Sp(2g,Z)
[Bo] imply that for any subgroup
of finite index:
H*(F,~)
~ ~[C2, C6, Cl0 .... ] , in degrees
~ g-2
In particular: H2(~,~)
This
~ H2(Sp(2g,~) ,~) ~ ~
if
g a 4 .
In parti-
355
An immediate
corollary*
is:
Theorem 1.5 C~ore~ £%~x|): Pie(A 0)~ ~ where
1
is the line bundle on
~ A° g
~.I
) ,'[ ~
)
defined by the c o - c y c l e
det (C~+D) . Corollary
1.6:
where
is the divisor class of the boundary
6
Pic'~( g1)'~.l~
In terms of these generators,
Proposition
1.7.
+ ~.~
a standard
A.
result is:
K (i),0 - (g+l)l-~. g
For a proof,
see for instance T a i [ T]I,§J •
Another
fairly
standard result that we need is: Proposition automorphism
1.8.
Let
group is
(B,E) be a (g-l)-dimension (~i).
Consider
*
If
Ag ~
~(b)
= the pair
> ~(I) ,0 g
(G,D) constructed
is a smooth compactification
of
res
plus
H 2~0g,~)
----+ H2(~g'~-) ....~
~ H 2(~g,~_) ~ ~
.
from
(B,E) with
_-~ 0g , then use:
o0 *~i
whose
the 2-1 map
~: (B_B2) defined by
p.p.a.v,
H2(~0'~)
356
gluein~ via b.
Then
@*CO~cl),0 C~))
0 B (-2_~).
~
g Proof:
Let's construct over B the family of
up w i t h all possible b's. over
BxB,
trivial on
as the universal ~l-fibre
bundle,
To do this,
Let
P D P
serves
be the associated
and
p = P/(bl,b2,0) Then the p r o j e c t i o n
(bl,bl+b2,~).
on the first factor:
PI:
is the universal
let P be the Poincar~ bundle
Then P* = P-(O-section)
exB, B×e.
family of G's.
(G,D)'s made
P
~B
family of G's.
The deformation
theory of such
a G gives an exact sequence:
0
>HI(G,T0(~I~))~
>TI(G)
>H0(Sing
G, ~TI(o~))
II H 0 (B,N0®N)
where
N0,N ~
of G.
For one G, made up starting
completed
at ~
are the normal bundles
to the locus of double points
from a line bundle L over B,
and glued by translation
NO ® N
~ L @ Tb(L-I).
by
b £ B,
357
Note that L must be algebraically hence
N08N ~ ~ 0 B.
vector
space represents
point
(G,D).
N08N ~
Thus
equivalent
H0(B,N0®N ~) ~ k. the normal bundle
BxB
L -1 '
TbL-I*
This o n e - d i m e n s i o n a l to
Doing this now for the whole
is the line bundle on
to 0, hence
~ in
~
family
g
at the
~
>B,
given by
P 8 T*(p -I) where
T(x,y)
= (x,x+y) .
Then the normal bundle to
A, pulled back to this family,
is
pI,,(P®T*(p-I))
which
is the same as the restriction
6*(P-I), where
6(x) = (x,x).
is
0(22),
this proves
§2.
The divisor
NO
Andreotti-Mayer
of P®T*P -I to
Bxe,
i.e.,
Since P, along the diagonal of
the Proposition.
BxB
QED
and its class in Pic(A~l)). f [A-M]
defined
the important
subsets
Nk
in A : g I
(2.1)
N k = {(A,0) ISing 0 ~ ~
Andreotti that
and dim(Sing
and Mayer prove by using the Heat equation
NO ~
Ag, but it is not easy to estimate
0) ~ k} for
the dimension
N k in general.
Nowever,
and we must at
least check that none of the Nk, k [ I, have
codimension
1 components.
of
we are interested only in codimension
This
follows by an elaboration
of
1
358 Andreotti-Mayer's
Lemma
2.2.
greater
arguments
u s i n g the h e a t e q u a t i o n :
T h e co dimension_ of N 1 (hence of N2,N3,-.-)
i_nn Ag i s
than 1.
Proof:
We use the h e a t e q u a t i o n
3# (2~i) ( l + 6 a 6 ) a a
32# 6
3z ~z 6 (\
If the l e m m a w e r e hypersurface containing
false,
g(£)
we c o u l d
= 0 defined
r,, c~t ¢, ~
f i n d ^ ~, a s m o o t h a n a l y t i c
in a n e i g h b o r h o o d
~, and a v e c t o r - v a l u e d
of
~
and
function
~(~,t) 6 ~g
defined
in a n e i g h b o r h o o d
%~(f(~,t) ,~)
of
~
and for
[t I small,
such that
- 0 whenever
3 ~ (~(n,t),~) 3z k
We m a y a s s u m e algebraic
that for e a c h
curve
g = 2 and if abelian
C~ c A~.
of d i f f e r e n c e s
for no
~
Note
is g r e a t e r
that the a b e l i a n v a r i e t y
differences
~, t [
>f(~,t)
x-y,x,y 6C
x-y,x,y eC~,
near
~
A
= 0.
is p a r t of an
that the lemma
g > 3, t h e n the c o d i m e n s i o n
varieties
g(~)
1 < k < g
_= 0,
than i.
generates
is there a v e c t o r
A
A~
we can also a s s u m e
It f o l l o w s , hence for
~
if
of the locus of n o n - s i m p l e
Therefore
is simple.
generates
is o b v i o u s
~
that the set
the set of near
such that
~.
Therefore,
359
÷ ~(a-~)
.÷ ~f. ta'~-~ = 0,
=
all t.
We prove b~ induction on d that:
(*)d
If lel =
d, then
Since
(z,~) does not vanish identically as a function of z, this
is a contradiction.
<
~ -h (f(~,t),~) ~z~1...~z~g / g
H 0
whenever g(~)= 0.
In fact, to prove this it will suffice to
apply: If
(**)
q(~,z)
satisfies
the heat equation and
~(~(~,t),~) E0
]
~n (~(~,t),~) ~0
J
~z k
whenever
g(~) = 0
whenever
g (~) = 0
then
~x--~z£
~,t) ,fl) E 0
to all the partial derivatives differentiate that if
of
~ in turn.
TO prove
the first relation with respect to
~k£
satisfies
tangent to the hypersurface 0 =
~k£~g/~k~(~)
=
We find
= 0, then
~+e~
is
g(~) = 0, hence
~ (f (~+e~, t) ,~+e~) ~fk
--
~.
(**),
~
b z--q ~n (fc ÷ ' t),~)._~ab.~ab + a ! b ~ a b ( ( ~ , t ) , ~ ) e ~2___q___~(~ (~, t ) ,~) 4~i a,b [ ~Za~Z b "~ab "
Web }
360
Therefore
~2~ ~Za~Z b (~(~,t)
with
some
f a c t o r ~,
differentiate
the
~g ,~) = ~(~,t) - (l+~ab) - ~ a b ( ~ )
for all
~
near
second relation
~, all small in
(**) w i t h
t.
Now
respect
to t.
We find:
~2q b[ ~Za~Zb
for all a,
If
~(~,t)
~0
substitution
w h e n g(D)
= 0, we are done.
whenever
g(a) = 0.
If not, we find by
that
for all a,
i.e.
~fb
(~(~,t) ,~) -~--~--(~,t) E 0
~fb (~)-~--{--(~,t) ~ 0
~(l+6ab) b
whenever
g(~)
= 0,
r +
~f (c(a)-~T)
(***)
= 0
where c(a) b =
For some a,
c(a)
But we saw that
(***)
In the o t h e r P r o p o s i t i o n Z.3
~ 0
since
(1+6 • )~-~g--(~). aD 0~ab
g(~)
= 0
did not occur,
direction, (Beauviile) :
so thus c o m p l e t e s
Beauville NO
is a s m o o t h h y p e r s u r f a c e .
[6], R e m a r k
has c o d i m e n s i o n
The r e s u l t is s t a t e d only for g = 4; h o w e v e r w i t h o u t any m o d i f i c a t i o n for all g.
the proof.
7.7 p r o v e d * : 1 -in -
Ag .
the a r g u m e n t w o r k s
361
His proof
also uses
an e l a b o r a t i o n
Mayer - - in this case equations
for the
could be p r o v e n without
their
N k.
technique
of A n d r e o t t i -
"explicit"
that this P r o p o s i t i o n
principles,
but I don't
one could
have e x c l u d e d
information,
that some c o m p o n e n t
for d e r i v i n g
(It m i g h t be thought
from general
specific
of the techniques
of some Nk,
k > i, was
see how~ the p o s s i b i l i t i e s
not in the closure
of
N0-NI.) We w a n t
now to c o n s i d e r
and to give m u l t i p l i c i t i e s would over
like
to use the
~(I). g
exist.
that over
U
~(i) g
there
family"
even g e n e r i c a l l y
group of order
However,
admits
are
N0
of
to its components.
"universal
However,
an a u t o m o r p h i s m
the closure
flat,
of pairs
~(i) g '
U
we
(A,@), (G,D)
these pairs
a "covering"
in
To do this,
2, so a u n i v e r s a l
proper
NO
still have
family
~
need
> ~(i) g
not such
families
Da c Ge
U
consisting and such fibre points
of a b e l i a n
varieties
that p is locally
(Gs,Ds). of
~
N
double points deformation
Outside U , G~
the u n i v e r s a l
~ N U
, G
itself will
of the fibres,
space:
and rank
p will
1 degenerations deformation
will be smooth still be smooth, look
like
thereof,
space over
of its U
; over
but at the
the u n i v e r s a l
local
362
~
~
~ [[Zl,Z~,Z2,-'-,Zg_l,t2,''',tg(g+l)/2]]
^
OTj
----~ ~ [[tl,t2,...,tg(g+l)/2]
]
(~
!
t I = Zl.Z 1 •
On
Ge' define
the subsheaf
of the tangent
sheaf
T
vert
to be
the kernel: 0
Note that the fibres,
> Tvert
% e r t is locally Tvert
> p*T U
e
free of rank g
is spanned by
Using a local equation T as derivations,
> T~
~ = 0
e
(at double
Zl~/~Zl-ZlS/~Zl,
of
points
~/~z2,''',~/~Zg).
De, and interpreting
sections
define:
Tvert
~
D I
> ~
OCD
)i/0 (independent
D6/~
of 6).
Let Singvert D
Thus
Singvert D e
= subscheme
is defined
codimension at most g.
of
De
where
e
locally by g equations
Set-theoretically:
of
is zero.
and has
of
363
(2.~)
p(Singvert De)
= set of points whose i) fibre is
fibres are of 3 types
(A,@), A abelian variety,
and 2) fibre is
0
singular
(G,D) and D has a singularity in G
3) fibre is
(G,D)
and the divisor
= D. (G-G)
TO see this at fibres of type
on G-G is singular.
(G,D), at points of G-G,
expand
6
!
in a power series in
Sin~ert D
in
Zl,Zl,Z2,.--,Zg,
t's:
then the origin
lies
if and only if
v
6 (Zl,Z I, ziz j (2 <_ i,j < g), t i)
i.e.,
if and only if ~ = 0 is singular
p(Singvert D ) patch together see shortly
that
The sets
~ [[z2,---,Zg]].
into a subset
~0
of
(We shall
~(i). g
~0 = N0")
Let us work out which G
in
,
(G,D) arise in cases
(2) and
(3).
Let
be the extension: 0
Then
G-G ~ B
> •
and
at the beginning
D. (G-G)
>G
case
>B
>0.
is the theta divisor of B, called
of this section.
the natural projection, AS for case
m
Thus if
(3) contributes
(2), if translation by
~: A
> Ag_l
~-I(N0(Ag_I))
is to
b 6 B is used in glueing
together G, then a local equation of D at any point of G is of the form f(x,z)
= ~p (x) + z. 6p+ b (x+b)
E
~0"
364
Here
6~ (resp.
which and
define z
6p+b ) are local
the n o n - z e r o
is a v e r t i c a l
G ~ ~ m x B. Taking
functions
section
coordinate
of
@B(H)
equation
(1.13) if we want.)
P, P+b 6 E
and
either
Looking that
z 6~*
at p o i n t s
N0
not a l r e a d y
the set of p a i r s
H '~ b
are t a n g e n t
YB:
be the
"Gauss map"
Tp, E, as a p o i n t
yB(P)
abelian
(B,E) w i t h
variety
c(B,~)
(G,D)
in case
to e a c h Then
= yB(P+b).
such that If
E
H
~
([n~
x such that
somewhere.
(2_0@) :
(G,D)
let
polarized
we may d e f i n e
= locus of p o i n t s
[U
is
is smooth,
for any p r i n c i p a l l y
x-y, w h e r e
~on~
B
E b are t a n g e n t at P
= locus of p o i n t s
T h e n in the d e s c r i p t i o n
H c
P 6 ~, the t a n g e n t p l a n e
Thus
smooth
tangent
(3), this shows
> pg-2
~(To,B).
if and o n l y if
covered
somewhere.
E
associating
of
_~ has the same tangent plane
at P,P+b, or is singular at both pts.
(G,D)
contains
s m o o t h and
(resp. P+b),
of f, we see that:
f(x,z) = 0 is sing~lar at x = P, some
near P
(resp. P+b)
on G in a local s p l i t t i n g
(We m a y use the a n a l y t i c
derivatives
on B near p
YB(X) E'Ex
= yB(y) are
365
Next,
the method of Andreotti-Mayer-Beauville
rank 1 degenerations, abelian varieties 1201, i.e.,
to prove
A, their
explicitly
that
technique
by the theta
(z,~) =
~
~0
extends
is a divisor.
to
For
is to map A to ~ 2g-I
by
functions
e 2zit(n+B)~(n+B)+4~it(n+~)'z
n Zg Call this They define
~:
A •
> ~ 2g-1 .
a linear
subspace
L~ c ~ 2g-I
@u(o,~2) . x
of codimension
g+l by
= o
(2.5) ~2J-~B (0,~)'X~ = 0 ~z
i<
.2
i<_g
1
and prove
~-I(L~)
(2.6)
= Sing
8,
hence (A,@) £ N O
Now if
Im
~4~
>~,
<
> L~ (~ ~(A)
<
> Chow form of ~(A) varieties at Pl~cMer Coord of L~
the limit of ~(A)
# 9~
is 9~(G), where
9~
is
defined by the 2 g "theta functions" (z(1)
~(i)) '
u~
+ u2~ (z(1)+~,~(1)) ~ (I) + ~1 , ~ (i) )
[ .I"
~6
1 2zg-i/zg-i
366
(where,
as above,
G is a ~ -bundle over B, m
B, G is glued via on ~m ) .
~(i) = period matrix of
~, z (I) is the coordinate
on B, u the coordinate
The basic theta identity on which the proof of
(2.~)
is
based becomes
[{)(x+y) +uw 0(x+y+~)] • [0(x-y) +U0(x_y+~) ] = (2.'7)
E% ~x)+u2% ~x+b>1-~ I~)+~% ~x9 .r ~ ( y 9 6 % ~- ~)I
X ~ 6 ~ g - i / ~ g -I
The limit of
L~
is the linear space
[ ,~(0,~ (1)).x + 2[ ~ (y,n
(1)
).Y
=
0
2% (2.~) 1
1
Z% (2,~ (I) )'% (The last equation comes to
w~Sw;
because,
from the 2n~d derivative
these equations in passing
T~en it follows
= 0
of
(2. 7 ) with respect
are not the exact analogs of the
to the limit, we have renormalized
from
(2.~)
(2.
the origin.)
exactly as in A n d r e o t t i - M a y e r - B e a u v i l l e
that ~-I(L~)
=
singularities (
of D in G plus singularities
of D" (G-G)
in G-G
)
hence Chow form of ~(G) Zero at L~ This proves
that
is~ /
~0 is a divisor.
(~,D) 6
~o.
367
On the other hand, it is clear that for all B, c(B,E) ~ B for generic B,E the closure
is smooth:
N0
of
NO .
always a divisor in B.
hence
~0 A ~ ~ ~ .
Thus
~0
Incidentally, this proves that
and
must be c(B,E) is
At the same time, we can now give
multiplicities to the components of N0"
I think the Andreotti-
Mayer-Beauville equation gives artificially large multiplicities, and want, instead, to assign multiplicities via the local description of
N0
in
U
as
set of points of
p(SingvertD ).
is finite over N~. N 0.
N~
be the maximal open
N 0 such that for all
P:
is dense in
Let
SingvertD~
Because
N1
~ (N0 N U e)
has codimension at least 2,
Then over N~ dim(SingvertD ) = dim N O
hence c o d i m ( S i n ~ r t D ~) = ~+l = # of equations defining SingvertD hence
Singvert D
is Cohen-Macauley.
Therefore, over
N0,
p,(OSin~ertD ) has a locally free resolution:
f 0
and
det f
> E1
> E0
gives a local equation for
Next, we want to break piece is
>p,(OSingv=rtD )
N0
N~ n D e ) ~
up into 2 pieces:
+0
~k,% ~E~,'~m9 ~n~,'~hcJ~% the first
368
if 0 is normalized to be symmetric about e, f
(2.q)
~null = i (A,0)
_
}
then 0 has a singularity at a point of order 2
It is easy to see that:
[ O~,D
for
Ag_l) ]
all
w h e r e we note
that
the
component:
"obvious"
(assuming
2B(H)
because
y(-x)
=
not of o r d e r
0
all
all m u l t i p l i c i t i e s
%ull
in the
the m a i n
Theorem
The d i v i s o r
IN 0 ]
=
[%~null]=
IN0 ]
=
contains
Z.
singular
We can now state
(2.10) :
x 6
has a s i n g u l a r i t i e s
2, it is also
N0 =
where
c(B,E)
too)
= {2x x 6 H}
y(x),
If a s y m m e t r i c
_= is s y m m e t r i c
result
(g+l) ! 2 +gl)l
at -x.
at a p o i n t Thus
x
N0 breaks
up:
--* + 2"N0
2 nd p i e c e
are d i v i s i b l e
by 2.
of this paper:
classes
of
N0'
#null'
(g+l) ! 6 12
2g-2(2g+l) I - 22g-5.~ (~+i) ! g! 2 g-3(2g+l) 4 + 2 -
I -
[
24
--* NO
are g i v e n by:
369
Here is a table for low degrees:
g
[~o ]
[~nu:L1 ]
3
181-26
181 - 2.5
0
4
841-108
681 - 84~
81-6
8
5
4801-606
2641 -32~
1081-146
7.71
6
3,2401-420~
1,0401 -1286
7
25,2001-3,3606
4,1281-512,5
Note that the figures imply
N; = ~
for
We also see that the divisor class of
N --* O
[N~]
sl~
1,1001-1466
7.53
10,5361-1,4248
7.40
g = 2,3
as is w e l l known.
is the same as that of
the J a c o b i a n locus for g = 4, c o n f i r m i n g B e a u v i l l e ' s results. last column,
"slope",
to the c o e f f i c i e n t of ratio for K,
Corollary
Proof:
Corollary
8.
As soon as this drops b e l o w the same
(9+1) , 12 " K~(1) g
=-
[!N0] + g! (g2-4g-17)l
Combine 1.7 and 2.10.
(2.12).
Proof:
refers to the ratio of the c o e f f i c i e n t of 1
A~is of general type:
(2.11).
If
Combine
The
g ~ 7,
Ag i~ of general type.
1.4 and 2.11.
370
§3.
Proof
of the T h e o r e m .
N o w h o w are we g o i n g [%ull ]
is i m m e d i a t e ,
o u t this d i v i s o r ,
f(~)
to p r o v e
the T h e o r e m ?
The
b e c a u s e we k n o w the m o d u l a r
formula
for
form that cuts
viz.:
=
T~g ~ [ b ] (0,~) a,bE~ /~ t(2a). (2b) even
where n~g~ e~i t (n+a) ~ (n+a)+2~i t (n+a) .b
'b%[a ](O,~q) =
b
Each
is a m o d u l a r
"even" p a i r s coefficient that if
a,b of
f o r m of w e i g h t
so f has w e i g h t
I.
1/2
2g-2- (2g+l),
On the o t h e r hand,
a I = 0, lim ~[
] = i, w h i l e
and there are
if
and this
Im ~ii
if
a I = ~,
of
A
2g-l(2g+l)
> ~[~]
is the
~ , we see is d i v i s i b l e
by e hence
it g o e s
t h e r e are
to zero.
2 2g-2
"even"
a,b
pairs
1 set a I = ~
a2,b2,-.-,ag,bg, force
The equation
to be even).
a,b
with
is aI
=
e 2~iQ11 1
~
Im ~11
>
~, h e n c e
(take any
a n d m a k e b I zero or o n e - h a l f Thus
f goes
to zero like
(e 2~i~I I)(22g-5)
when
= 0, and
the c o e f f i c i e n t
of
6.
to
371
It r e m a i n s coefficient
of
Proposition
3.1:
to p r o v e ~
the f o r m u l a
follows
for
[N0].
The v a l u e of the
from:
Let X
~
D
C
be a f a m i l ~ of p . p . a . v , theta divisor the ~ e n e r i c
Dt Dn
over a complete
curve
has o n l y a f i n i t e n u m b e r is smooth.
~:
L e t this
C
> A
g
C
such that e v e r y
of s i n g u l a r i t i e s
family define
and
the m o r p h i s m
.
Then
~*N0
H
(
(g+l)' 2 " + g!)~*l
(Note that such a f a m i l y e x i s t s in S a t a k e ' s
compactification,
The c o e f f i c i e n t
Proposition o_~f c(B,@)
3.2:
of
because
the w h o l e
6, on the o t h e r h a n d
Let
(A,0)
c (B,0)
together with Proposition
codim N 1 ~ 2 boundary follows
be a p . p . a . v .
is 9 i v e n by:
-
1.8.
(g+2) ! 6
+ torsion.
Then
and b e c a u s e
has c o d i m
~ 2).
from:
the d i v i s o r
class
372
To prove
3.1, we use the exact sequence
Tx/c
Ox(D)/O X
used to define m u l t i p l i c i t i e s
> ~Singvert D ® OX(~)
for
NO.
It follows
that
>0
Singvert~
is the scheme of zeroes of a section of
nX/C(D)®Ox°~ hence ~0*N0 = p.(Cg(~i/c(D)) . D)
But if
1 ~X/C'
1 E = p. (~X/C) , then the bundle
each fibre of definition
X over C, is isomorphic
of
to
being trivial on
P*E.
Moreover,
by
l,
~*l
= cI(E).
Thus ~*N 0 = p. (Cg(p*EOO~(D)). D ) = p.((Dg+Dg-l.cl(P*E)) .D ) = p.(D g+l)
Now on each fibre D is e g!~*(1).
To compute
Roch theorem to
and
+ p.(Dg)-cI(E)
(0 g) = g!, so the second term is
the first, we apply the G r o t h e n d i e c k - R i e m a n n -
OX(D).
Note that
p.(O X(D)) i R p.(O,(D))
~
OC
= (O) , i ~ i.
373
Thus 1 = ch(p, (Ox(~))) 1 = p,~hOx(9).Td(~x/c)) = p,(eD.p*(l-C1~E----~))),
In c o d i m e n s i o n
1 on C,
this
mod
torsion.
says ~g+l
0 = p.((g~.,)
el(E) 2
"P*(
. )
or p.(Dg+l)
This
proves
and
then
divisor
3.2,
of the Theorem
classes.
(C.c(B,@)).
(g+l) 2
it s u f f i c e s
2 divisors. 2.10 w i l l Let
Consider
Namely, imply
C c A
-i
x = y'-y,
(@)
mod
torsion.
is the
where
the n u m e r i c a l
this w i l l
Prop.
be any
prove
Theorem
3.2 as an e q u a l i t y
curve.
We
2.10,
of
shall
calculate
x+y
6 @ , i.e.,
the m a p
m(x,y)
m
Cl(E)
to e s t a b l i s h
C × 0
Then
!
3.1.
To p r o v e equi~ence
_
locus
m = x + y
of p a i r s
x £ C,y,y'
>A
6 0.
.
(x,y)
The
where
differential
of
m
gives
a map dm:
P 2*T 0
8
Om_l@
>
T A ® 0@
• N0, A
us
374
whose
zeroes
X = y'-y, above
are e x a c t l y
the p o i n t s
Y,Y' 6 0 , but also
dm
can be t h o u g h t .
T
O
(x,y) Ty,,0,
=
of as a s e c t i o n 1
P2~0 0
m*
such that not only
(N0, A ) 0
i.e.,
x
6 ~-
is
Now the
of
(0 -I m (0)
hence * 1
(C.~) = C g _ l ( P 2 ~ 0 0 m*((0(O))
Let
01 = pt.x0,
equivalence)
on
be these
02 = m - 1 0 CX@.
@
(0m-10)"
divisor
classes
(mod n u m e r i c a l
Then ,
(C.~)
1
= Cg_l~P2~
0 0
(0(02)) .02 .
Using 0
>
(0(-0)/0(-20)
>
~" ~ 0
> O,
we see that c(~l)
=
(i-0)-i10
=
(1+(9 + (9 2 + . . . . . )
Io
Thus (C. D) =
81g-1 .02 + 0g-2.022 + ..... + 0g "
But now k g-k (81"82 )C×8 =
k+l. (m(e I )- e g - k ) A
=
((C ~ @k+l).@g-k)
if + is P o n t r y a g i n
product.
By s y m m e t r y
of
=
(C. (8 k+l $
=
(C. (k+l) (g-k) ( g - l ) ! @ ) A
@, this
is
0g-k)) A
Thus (C.D)
=
(C.8) (g-l) !
(k+l) (g-h) k=0
_
(g+2) ! (C 0) 6 " "
QED
375
References
[A-M]
Andreotti, A., and Mayer, A°, On the period relations for abelian integrals on algebraic curves, Ann. Scuola Norm. Pisa, 21 (1971).
[A-M-R-T]
Ash, A., et al, Smooth compactification of locally symmetirc varieties, Math-Sci Press, 53 Jordan Rd., Brookline, MA, 1975.
[H]
Beauville, A., Prym varieties and the Schottky problem, Inv. Math., 41 (1977), p. 149.
[Bo]
Borel, A., Stable real cohomology of arithmetic groups II, in Manifolds and Lie groups, Birkhauser-Boston, 1981.
[c]
Clemens, H., Double solids,
[D]
Donagi,
[FI]
Freitag, E., Die Kodairadimension yon K6rpern automorpher Funktionen, J. reine angew. Math., (1977), p. 162.
to appear.
R., The unirationality of A 5
, to appear. 296
[F2]
Freitag, E., Der KSrper der Siegelschen Modulfunktionen, Abh. Math. Semi Hamburge, 47 (1978).
[H-M]
Harris, J. and Mumford, D., On the Kodaira dimension of the moduli space of curves, to appear in Inv. Math.
[I]
Igusa, J.-I., A desingularization problem in the theory of Siegel modular functions, Math. Annalen, 168 (1967), p. 228.
[M]
Mumford, D., Analytic construction of degenerating abelian varieties, Comp. Math., 24 (1972), p. 239.
[N]
Namikawa, A new compactification of the Siegel space and degeneration of abelian varieties, Math. Ann., 221 (1976).
[s]
Stillman, M., Ph.D. Thesis, Harvard University,
[T]
Tai, Y.-S., On the Kodaira dimensions of the moduli space of abelian varieties, to appear Inv. Math.
1983.
GENERALIZED
HILBERT
FUNCTIONS Ferruccio
OF
COHEN-MACAULAY
VARIETIES
Orecchia
Istituto di Matematica Universit~ di Napoli NAPOLI-ITALY
INTRODUCTION.
Let
nsion
maximal homogeneous ideal
d
and
R = kIXo,...,Xrl/ I
be a graded algebra over a field M
. Let
k
of dime ~
H~(n) = d_~(~. ~m M n ~/.. n +I) and
o PR(n),
n • Z , be respectively ~h~ Hilbert function and Hilbert polynomial of the ring
R .
In many papers (see for example [G.M.] , [G.O.]and [03]) the relations between the structure of the Hilbert function of ideal
I
have been studied. If
R
R
and the degree of the forms generating the
is Cohen-Macaulay these relations are much more
strict; for example Schenz~l ( see [Sc] ) proved that if is the index of regulm~ity of
R
and
t
is the least degree of a form of
m.+ d ~ t . Unfortunately few sufficient conditions for known.
m + d Zt
minimal generating set of R
R
I
then
to be Cohen-Macaulay are
In this paper first we improve the previous result of Schenzel by showing
that the inequality
for
m : Max {n I ~(n)~P~(n)~I
I
to be Cohen-Macaulay
Cohen-Macaulay ring (
holds also if
t
is ~he highest degree of a form in a
( see thm. 1.5 ). Then we give a sufficient condition when
R
is the associated graded ring of a local
in particular the tangent cone of a variety at a Cohen-Ma-
caulay singularity). For this purpouse we have to introduce the notion of generalized H_ilbert functions and
for
function
i
~
~(n)
, i • Z , which for i positive are successive Sums of
negative are successive differences of -d(n)
is maximal then
This work was partially supported by
R
is
C.N.R.
~(n)
~(n)
. We prove that if the
Cohen-Macaulay ( thm. 3.1)
The
377
notion of maximal Hilbert function , which we give for ary tends the one given in IO 3] for the one-dimensional Macaulay ring as contained ~-d(n)
~(n)
, unifies and ex-
case and that of extremal Cohen-
in ~Se] • Furthermore large classes of varieties have
maximal; for exa~01e: i) a set
of
~r
points in generic position in
2)
curves locally requiring large numbers of generators ( in particular the famous Macaulay curves [MI ) , 3) surfaces with ratioml
singularities , 4) determinantal va-
rieties . Hence as a oormequence of our results 2) and
3)
case
we
get that if
3.3
of [O1].
i.
GENEBALIZED
have Cohen-Macaulay tangent cones . In particular in the o~-dimensional ~(n) is maximal then
HIT.R~T
FONCTIONS .
is a ~raded al~ebra over a field Let
dim R : d
functions of
let if
neZ ie Z
~(n)
and R
relative to
: if
the functions
a
= e(R) A I O
P~(n) = 0 H
o
is Cohen-Macaulay
. This extends thm.
From now on we assume that
of maximal homogeneous ideal
R: k[Xo,...,Xr]/l M : (Xo,...,x r) .
as follows:
~(n)
~+l(n-l)
(n)
d~(Mn/Mn+l)
; if
n
, ~(n)
=0.
are given by the relations: ( then
for
~(n)
n>>O
: ~. ~-l(j) j<_n
al(dn2) + ...+ ad_ I
is the multiplicity
) .
O
is a numerical polynomial PR(n)
called the Hilbert polynomial ( of
PR(n) : ao(d~_l) +_
and
M
riCO, ~ ( n ) =
: ~+l(n)-
d-i
k
R
emdim R = d ~ ( M / M 2) = r+l . We define the generalized Hilbert
It is well known that degree
we get that the varieties of type
where
R
relative to M ).
a.e Z 1
( or degree) of
of
Then , if d >0:
'
i :0 .... , d-I
R
in
M . If
d = 0 , then
378
Hence , for
4 for
n>>O , we have :
R .n+i. .n+i. (n) = P (n) = ao(d+i_ I) + al(d+i_ 2) + ...+ ad+i_ 1 , i > - d and i_<-d
.
DEFINITION 1.1. ~
: Max { n e Z
The index of resularity of ( the function ) ~(n) is the integer l~(n)~
index of regularity of
PR(n)} + I . The integer
~
will be simply called the
R
1.2. The following equality holds
PROOF.
~
= ~+i
+ i
Follows 9maediately from the equalities :
~+l(n)
-
~+l(n-l)
=
~(n),
i+l
PR
i+l,
(n) - P R
= PR(n)
(n-l)
R~NARK. The notion of index of regularity of the ring ~(n)
a)
for which
b)
if
If
( i.e. of the function=
d = d i m R : 0 , the index of regularity of
M n : 0 , i.e. the r~ipotency degree of
d : i
the index of regularity of
R
is
negative as the following example shows . Let ~(n)
: 2(~) + 3n + i ,
for
nAl
R
R
is the least integer
( see [S] , pg. 79 ) ;
Min {m~ N l~(n) : e(R)}, for n A m
i.e. the index of stability as defined in [L I ; c)
Then
R
, neZ
) has been introduced by Schenzel in [ScJ
EXAMPLES. n
(n) : P~(n)=O
the index of regularity can be
R : k [X,Y,Z,T ] / ( X2+y3+z4+~
.
Now the following key result holds :
THEOREM 1.3. Let
x
be a homogeneous element of
M . For every
n~O
:
) •
379
~/xR(n) - ~ ( n ) and only if
: d~(
~(n)
= ~/xR(n) for any
@emeous elements of Xo,...,x
Mn+I:xR/M n ). Then x is a non-zero divisor of degree 1 iff
M
form an
and
n_>O . More generally,if
A = R /(Xo,..~Xs) , ~ ( n )
R-sequence of elements of degree
= H:(n)
Xo,...,x s are homoif and only if
1 .
S
PROOF.
We have
: d~(
~+i
~(n) -
+ RxPMn+l ) : d ~ (
R--xR/(Mn+inRx) --0 Hence
H~/XR(n) : d ~ k ( R/ M n+l ) - dimk( R /( Mn+I+Rx ))= RX/(Mn+
induced by multiplication by
d ~ k ( Rx/( Mn+inRx )): d ~ (
:~/xR(n) - ~ ( n )
) . Now the homomorphism
+ ~(n-l)
has kernel
Ai: R/(Xo,...,Xi_l) ( A ° : R ),
, by the first part of the theorem the image
non-zero divisor ~ if and only if
~A
(n) : i/(xi)
: H~(n)
:
: - d ~ k ( R/( Mn+l: xR )) + d ~ k ( R/M n) : d~k((Mn+l:xR)~ n]
x. of i
~(n)
( M n+l : xR) .
R/(M n+l : xR )). Then : ~/xR(n) -~(n)
Now we prove the second part of the statement . If i : O,...,s
x
is equivalent to
~ i
H ° (n) A.I
(n) = H i+l (n) , for any Ai+l
A. is a
1
for any '
x. in
i
n>O -
But "
i = O,...,s-i .
REMARK . Actually , in the previous theorem , x is a non-zero divisor if ard only if ~(n)
: £/xR
(n)
fl
(1.4)
If
k
for
n
large enough , but we will not use this weaker statement .
is infinite and
Xo,...,x s_lof elements of degree
B : R/( Xo,... ,Xs_ I)
Now let
s
t
i . In this
Xo,...,Xs_ I _ _in
are the classes of
e(B):e(R) if
depth R : s i d there is always an
, s <_ d
case we can assume that
R-sequence Xo,...,Xs_ I
R : k[ Xo,...,Xr] / I . The ring
, has • dim B : d -s , emdim B = emdim R - s : r + i -s,
, e(B) : d ~ k (B)
if
s : d
( see IS ]).
be the highest degree of a form belonging to a minimal system of
380
generators of
THEOREm4 1.5. of
R
then
PROOF.
I
:
If
R
is a Cohen-Macaulay ring and
m + dAt
m:~
.
By the standard trick of making the flat change of rings
( u indeterminate ) , if necessary , we can assume ( because
R
k
i .
Let
see thm 1.3) • Then , if
m
A : R/(Xo,...,Xd_l)
. We have
is the index of regularity of
(fl,...,<)
:'[
( the
fl"'"f~
where
h
f~(fl,...,f) has degree
s : 0 . Let
. Then
n-i . Then
-f c lhx
But
h : 0
C
~(n)=HA(n)
HA(n)
( see
, m + d is lemma
1.2).
Xo,...,X s ,
s
) . It is
be the least degree of a form f e l m = ( ..., ) . Hence f - Z h . f . = hX , ~I' }m i=l i i o
: 0 in
R . But
. Contradiction
x
is a nca zero divisor of
R
O
.
Tbms we can prove the theorem for
( Xo,...,Xr)mc I and then
I
d : 0 .
is generated by forms of
< m .
REMARK .
OPEN
hel
a~(m) = 0 clearly gives
degree
thm
i.e.
: H~(n)
~(n)
n
O
Hence
Xo,...,Xd_ I
: I if and only if
"-" denote the classes modulo
enough %o prove the claim for such that
I , (fl'"" 'f)
depth R : d
R-sequence
the index of regularity ( i.e. the nilpotency degree ) of Now we claim that , for
R--R(u):RIuI~Iul
infinite . Now
is Cohen-Macaulay ) , hence there exists an
of elements of degree
(
is the index of regularity
The previous theorem Lvproves the following result of
and cor.
QUESTION.
t= m + d .
i ) : if
l
Characterize
is the least degree of a form of
geometrically
Schenzel ( see [ S 1 , I
then
t : m + 2
•
the varieties satysfying the equality
Ciro Ciliberto has informed us that : Cohen-Macaulay curves of
homogeneous coordinate ring has
m + dkg
are linked to plane curves .
1~3 whose
381
2.
MAXIMAL
n>O
HILBERT
, and so for
i>O
(i) = r + i + I =
for any
,
O
DEFINITION 2.1. aN
n
,
say that
If
r + I = endim R , on has
~(n)<_(n+r+i r+i ) " Further , for
ieZ
(r+i+l-) r+i and it is easily checked that if ~
then
(m)<
(m+r+i) r+i
The function
l
FUNCTIONS .
(n)
~ ( n ) _< (n+r)r
, -r<-d
,
.n+r+i. (n) = ~ r+i )
Hence we can give the following :
_ ( i >-d ) is maximal if
= index of regularity of
(n) . If
~
(n+r+i) r+i
(n) =
(n)
for
is maximal we will
has maximal HAlbert function .
From now on we denote with
m
the index of regularity of the graded algebra
R = k,,IXo,...,Xrl/ I .
REMARKS . i) sense .
2)
For If
m
i < -d
the notion of maximal HAlbert function has clearly no
is the index of regularity of
always maximal because its index of regularity is ra~ximal implies points in ~ r
~+l(n) then
in generic position
R
maximal . 4)
If
R
R
the function
m-(m-2) = 2 . 3)
~R-2(n) is ~(n)
is the homogeneous coordinate ring of
has maximal HAlbert function if and only if these points are
( see [02] ,
thm 3.7
and
def. 3.8 ~.
There is an easy way of characterizing the maximality of the function
LE~NA 2.2.
Let
if and only if
l
be the least degree of a form of
I zm-i .
I
Then
(n)
~(n)
.
is maximal
382
PROOF.
By definition
~
(n) is maximal if
(n) :
but this is equivalent to say that ~(n):(n+r) of degree
Then if generating
(2.3)
n
i n<~:m-i
.
is Cohen-Macaulay and
q
in
R I
I
, for
(n+r+i) r+i
( for any
n
,
l
h~
)
(lenne 1.2 ) i.e. there are no forms
is the degree of a form of a minimal set
, on has :
~(n)
is maximal if and only if
m-i_
In particular :
PROPOSITION 2.4.
l_~f R
is Cohen-Macaualy the following conditions are equivalent :
i)
~d(n) is maximal,
2)
I
is generated by forms of degree
3)
I
is generated by .
m + r
(m+r) r-d
m + d
linearly independent forms of degree
m + d and ---:
.
e(R) : (~_d+l) .
PROOF.
i)<=~2) . ( See lenma 2.2 ) . 2 ) 0 3 )
the classes If
Now
Xo,...,Xd_ I
of
. We can assume
k
infinite and that
Xo,...Xd_ I form a regular sequence in
R : k[Xo,...,X~LI.
B : R/(Xo,...,Xd_l) , dim B : 0 , emdim B : r +i- d , e(R) : d ~ ( B ) m+d-i d~k(B ) : ~ ~(n) n:0
I mod Xo,... ,Xd_ I and we have
v(1)
: v(~)
v(
m+d-i ,n+r-d t m+r : ~ ( r-d ) : ~r-d+l ) " Further if n:O )
~
(see (1.4)) .
denotes the ideal
denotes the minimal number of homogeneous generators ,
( see the proof of thin. 1.5 ) • But
B ~ k [ Xd_l,...,Xrl /
383
a~
then
~(m+d)=O
which implies
~=(Xd_l,...,Xr)m+d
and so
v(I)=v(~)=(m+d ) •
2)
of prop. 2.4 have been
EXAMPLES . The Cohen-Macaulay rings satisfying condition
called extremal rin~s by Schenzel ( see [Sc] ) . Examples of extremal rings are the following .
I)
Let
nat@s over a field maximal minors of 2)
Let
~A,M)
X=(X..) , l < i < n ij k
X
and let . Then
isj~m
A: k[X] . Let
,
l(n,m)
R: A/l(n,m) is e~tremal
n<m
R : G(A)
is a
, a matrix of indetermi-
be the ideal generated by the (
see [Sc] , n ° 4 )
be a local Cohen-Macaulay ring of dimension
Then the associated graded ring [S] ,
,
d
and
;
emdim A=e(A)+d-l.
Cohen-Macaulay extremal ring
thm 3.10 ) • This is the case of a local ring
( see
A of a ration~l surface singu-
larity.
PROPOSITION 2.5.
Let
R
be Cohen-Macaulay and
Xo, .. .,Xd_ 2
__an R-sequence • Let
B = R/(Xo,...,Xd_ 2) . The following conditions are equivalent :
i)
~
2)
I
3)
~(n)
-d(n)
PROOF.
is maximal,
is ~enerated by forms of degree : Min {(n+r' r' ) , el
See
~(n)~e(A)
REMARK . Let
, where
m+d-i
and
e : e(B):e(R)
(2.3) , thm 1.3 and recall that if , for any
V
n~ Z
B
m+d , and
r'+l: em~im B : r-d+l
is Cohen-Macaulay of dimension
~, ( se [S] , ch 3 , thin 1.1 ).
be a projectively Cohen-Macaulay variety of dimension
over a n algebraically closed field . In this case the maximality of characterized by section
VnS
3)
i
of prop.2.5 : V
has
, with a linear subspace
consists of points in generic position .
S
~-d(n)
d
~-d(n)
in ~ r can be
maximal if and only if a generic
of projective dimension
r-d+l of ~ r ,
384
It is then clear the importance of finding sufficient conditions for
R
to be
Cohen-Macaulay . Unfortunately the maximality of the Hilbert function of
R
is not
enough as the following examples show :
EXAMPLE i . ( Hartshorne - Hirshowitz ) . If
V
of
then
e
Hence
skew lines in general position in ~°r ~(n)
is a projective variety consisting ~ ( n ) = Fain { (n+r) , en+e}(see[H])
is maximal , but it is well known that
R
is not Cohen-Macaulay ( see
[a.w.]).
The following example was shown us by Ciro Ciliberto :
EXAMPLE 2).
Let
geneous ideal in PI,...,P6 . If
PI'"'P6
I
6
2 points of a conic in ]Pk and let
k[X,Y,Z] generated by the forms of degree R : k [X,Y,Z] /I
~ ( n ) : 6 : e(R) , n 2 2 othe.~wise
be
. Then
it is easily checked that
~l(n)
is maximal and
R
would coincide with the ideal of points
3. THE CASE OF ASSOCIATED GRADEDALGEBRAS R = G(A) : Q (Mn/Mn+l)
3
I
be the homo-
vanishing on
dim R = 1 and
~(I)=3,
is not Cohen-Macaulay ;
PI"'"P6
"
From now on we aSsume that
is the associated graded ring , at the maximal ideal
M
, of
R
is
n~o
a local Cohen-Macaulay rin$
(A,M) , of dimension
emdim A = r+l and rm~itiplicity
d
, embedding dimension
e : e(A) .
In this case the maximality of the Hilbert function Cohen-Macaulay as we will show .
1 - d ( n ) gives that
3~
All the definitions regarding Hilbert functions of be extended to
A
= ~(n)
given in section
2
can
In fact :
HA(n) = d ~ ( M n / M n+l) = ~ ( n )
~(n)
R
,
i~z
,
•
th~n:
e(A) = e(R) , e m d i m A
= emdimR
, dimA
= dimR
.
Further a result corresponding to thm. 1.3 holds :
THEORY4 3.1. Let Then if
x~ M
Xo,...×se M
For every and
and only if the classes de~ree
1
n~O
,
B = A/(Xo,...x s) in __
Xo,...x s
In particular ,
HA/xA (n)-
(n) = d i m k ( M n + l : x R / ~ ) .
, H~(n)~ 4(n)
R = G(A)
i_~f s = d-i ,
form an
~(n)
and equality holds if
R-sequence of elements of
= H~(n)
if and only if
R:G(A)
is Cohen-Macaulay .
PROOF.
The same as that of
THEORKM 3.2.
PROOF.
Hl-d(n)
is maximal then
By passing to the ring
there is an let
If
thin . 1.3 •
A-sequence
A(u)
x o,...,xd_ I
B = A/(x ° .... Xd_ I) . We have
R = G(A)
is Cohen-Macaulay .
we may assume that
k = A/M is infinite ; then
of elements whose classes in
e = e(A) = d ~ k ( B )
( see IS] ,
emdim B = r'+l : emdim (A) - d . Further if
N
B ,
n>>O . Then , for
~(n)
: d ~ ( B / N n) : d ~ k ( B )
(n)-<Min{(n+r'r' ) , e } . Now
HA
(n)>_~(n)
, for any
: e , for
R
have degree i;
thm 3.4 ) and
is the maximal ideal of the local ring
Hl-d(n) __ M i n { ( ,n+r') r , e}
n>_0 • By adding , we get
n~0
,
( s e e def. 2.1 ) and so
(n~ ->
(n)
. Since the
386
;
reverse eq~lity always holds we have
ard
R
is Cohen-Macaulay by
thm. 3.1.
REMARK.
In the one-dimensional case thm. 3.2 extends thm. 3-3 of [01].
It is not true that if l-d then
G(A)
n~3
Then
has maximal Hilbert function of any degree greater than
is Cohem-Macaulay as the following exar~ple shows:
EXAMPLE. Let for
A
A = k [t4-1, t ( t 4 - 1 ) , t 3 ( t 4 - 1 ) 2] then
( see [02] , example l.b ) .
H~(n)
H~(n) : P ~ ( j
has maximal Hilbert function . Fur.ther G(A)
This provides an exa~ole for
C = A [XI,...,Xd_I]
C-sequence and so
i Hc(n) : ~+d-l(n)
C
: 4n-I , for
,
d~2
. Thus
. Then , if
is not Cohen-Macaulay since
n~2
.
is not
d m 1 . To have an example for any
consider the ring
maximal but
Hence
H~(1) : H~(2) : 3 , H~(n) = 4,
d
.
it suffices to
dim R = d , XI,.. ~Xd_ 1 is a
i>l-d,i.e~
i+d-l>O
A = C/(XI,...,Xd_ I)
H~(n)
is
is not Cohen-Macau-
lay .
Now we assume that a~iideal
I
;
let
A : B/I
is a quotient of a regular local ring
I" is the homogeneous ideal of
element~ of
I
The informatlons
rmmber of the forms generating If
f
is an element of
feN n .
Let
v(1)
modulo
emdim (A) : r+l then:
G(A) = k [ X o , . . . , X r l /
where
(B,N)
B
G(A) of
I"
generat@d by the leading forms of the
prop. 2.4 , 2.5
I" extend to
we call degree of
I
on the degree anti the
, using the following crucial lermm. f
the least integer
be the least number of generators of the ideal
nAO I
:
such that
387
Let
n
be a positive integer . l_ff I
and
n+l , then
I
is @enerated by elements of degree
N(I)
: v(I')
LEMMA 3.3.
PROOF.
(i)
n
and
n+l ; further
.
( See [03] , lerm~ i ) .
THEORN~ 3.4. m
is generated by forms of de@ree
,
If
is
Cohen-Macaulay of dimension
d
and index of resularity
the following conditions are equivalent:
HAd(n)
(ii)
A = B/I
I
is maximal
is generated by elements of de$ree ~ m+d , I is minimally generated by
Macaulay and
e(A) = (rm~l)
PROOF. (i)~=~(ii) elements of degree (ii) ~ ( i i i )
EXAMPLE.
A
r-d
elements of degree
prop. 2.4
m+d ,
prop. 2.4
I
and the remark that
if and only if such is
. Follows from
If
(m+r)
R = G(A)
THEOREM
3.5-
is generated by
and ler~na 5.3 •
is a local Cohen-Macaulay ring such that
er~im(A) = e(A) + d -i ( in HAd(n)
is maximal.
is Cohen-Macaulay :
The following conditions are equivalent :
(i)
H~-d(n)
(ii)
I
is maximal,
(iii)
I is senerated by elements of degree
is generated by elements of degree
Macaulay .
is Cohen-
I~
particular the local ring of a ratior~l surface singularity ) then Hence
G(A)
.
. Follows from m+d
•
m + d - i m + d - I , m + d
and
G(A) is Cohen-
388
PROOF .
(i)~=~(ii).
(See
lenmm
In the one-dimensional case
COROLLARY 3.6.
Let
sitive i n t e g e r
p
2.2).
thm 3.4
dim A : I then such that
I
(ii)~(iii)
.( See
thm. 3.2
and
(2.3)).
can be strenghthened:
HAl(n)
is maximal if and only if there is a po-
i s g e n e r a t e d by e l e m e n t s o f d e g r e e > p
an:l
e(A) = (p+r-l) r
PROOF. 3 ) ( r+l = and and
• (See thm. 3.4) • ~
emdim A ) and
) • By hypothesis
H~(n) : (n+r)r for
H~(p-l) = e(A) , ( this implies
H~(n) : Min{ (n+r)r , e(A) } is maximal , whence
H~(n) : e(A)
n
for
nap-i
G(A) is Cohen-Macaulay ( th~ 3.2)
p-i = Min {n l(n:r)~e(A)} is the index of regularity of
A . Nox apply thm 3.4.
REMARK . The previous corollary extends prop. 12 of [03.] and can be applied to various classes of curves locally requiring an arbitrary large number of generators . In fact the classical Mcaulay's examples ( see [M] ) , Moh's examples ( see [Mo] ) , Maurer's examples curves
( see [Ma] ) have all maximal Hilbert function of degree
of the form
is generated by
Spec (A) , where
elements of degree
P is minimally generated by [Mo]
anti [Ma]
p+l
A : k[X,Y,Z]loc/P
,
p . So corollary 3.6
-i . They are all
e(A) : (p~l)
and
P
plus thm. 3.4 prove that
elements as was claimed in [M] , and proved in
but furthermore we have that
G(A'~ is Cohen-Macaulay which was never
pointed out before .
OPEN QUESTION . Let ty . Suppose that We b~ve seen that if
A A
be the local ring at a singular point is Cohen-Macaulay and let
~-d(n) }%
is maximal then
V
V
p
of an affine varie-
be its tangent cone at
p .
is projectively Cohen-Macaulay
3~
(t~n 3.2 ) • In this case if an
R -sequence HA-d(n) : ~ ( n )
x o,...xd_ 2
is the homogeneous coordinate ring of
of elements of degree
(t~n 3.1) , and
following question . Let that
R
~(n)
Xo,...Xd_ 2
be an
dimension
V
B : A/(Xo,...Xd_2),
A-sequence of elements of degree
V
V
i
projectively Cohen-Macaulay ?
such O
, with a linear subspace of projective
r - d + i , consists of points in generic position is then
Cohen-Macaulay ? . In particular if of
Then, if
there is
is maximal . Hence it is natural to ask the
~ ( n ) is maximal; is then the tangent cone
Or , equivalently , if a generic section of
i
V
V
projectively
V is a curve and if a generic hyperplane section
consists of points in generic position , is then
V
projectively Cohen-Macaulay?.
390
R E F E R E N C E S
[G.M.]
A.V.
GERAMITA
[G.O .]
~n
A.V.
- C.
GERAMITA
R. H A T S H O R N E
WEIBEL
J.
LIPMAN
(1971)
[M]
F.S.
,
~
Droites en p o s i t i o n generale dans l ' e s p a -
,
( ~reprint ) , Stable ideals and Arf rings , Amer. J. Math. , 93 , 6 4 9 - 6 8 5 ,
,
The algebraic theory of modular systems , Cambridge
MACAULAY ,
[m]
J. MAURER ,
[Mo]
T.T. M O H
±
•
,
University
[Q]
t
On the Cohen-Macaulay and Buc~baam p r o p ~ t , Queen's Math. Preprint , No. 1 9 8 2 - 1 6 ,
,
- A. H I R S H O W I T Z
ee p r o j e e t i f [L]
of forms vanishing a t a f i n i t e Preprint No 1 9 8 1 - 5
Minimally generating ideals defining c ~ t t a i n J o ~ n a l of Algebra , 78 , 3 6 - 5 7 ( 1 9 8 2 ) ,
,
t i e s for union of l i n e s i n
[~]
•
A.V. GERAMITA - F. ORECCHIA ,
tangent eon~ [G .W.]
, The i d ~ l Queen's Math
- P. M A R O S C I A
s e t of points i n
,
1916
,
Eine variante der Moh-C~ven
( P~ep~nt ) ,
, On the unboundedness of g e n ~ a t o r s of orime i d e ~ i n power s e r i e s rings of three variables , J. Math. Soc. Japan , 26 , 7 2 2 - 7 3 4 , ( 1 9 7 4 ) ,
F. OBECCHIA , Ordinary s i n g ~ a r i t i ~ Bu//. , 24 , (4) , 4 2 3 - 4 3 1 , (1981),
of algebraic c ~ v e s , Canad. Math.
One-dimensional l o c a l rings with reduced associated graded ring and t h e i r H i l b ~ t t function , Manuscripta Math. , 32 , 3 9 1 - 4 0 5 , ( 1 9 8 0 ) ,
[0 2 ]
F. ORECCHIA ,
[03 ]
F. OBECCHIA ,
[s]
J.
[sc]
P. S C H E N Z E L
Maximal H i l b e r t functions of one-dimensional l o c ~ rings, Proceedings of the Trento Conference on Commutative Algebra , Lect. Notes i n Pure and applied Math. , Marc~ Dekker , 223 - 2 3 3 , 1983 ,
SALLY , N u m b e rof generators of ideals i n l o c a l rings , Lect. Notes i n Pure and AppSied Math. , Marc~ Dekker , 1978 ,
ge
,
Journal
,
Cber die f r e i e n auflSsungen extremaler Cohen-Macaulay t i n of A l g e b r a
,
64
,
No. i ,
93-101
, (1980)
,
SOME CURVES
IN~-
ARE SET-THEORETIC
Lorenzo
Robbiano
It is well known that a longstanding
COMPLETE
and Giuseppe
problem
INTERSECTIONS
Valla
in algebraic
geometry
is
3 whether
every connected projective
curve
in ~
k
is a set-theoretic
complete 3
intersection
(s.t.c.i.).
Let C 4 be the quartic
4
parametrically minimally deg
by
3
3
rational
in~k
given
4
X0=u , Xl=u t , X2=ut ,X3=t ; i t s d e f i n i n g i d e a l can be
generated
by four polynomials
FI,F2,F3,F 4 such that deg F2=2,
Fl=deg F3=deg F4=3 and t h r e e o f them, say F1,F2,F 4 d e f i n e C4 schema-
tically.
Further
(see
[6])
C
is a s.t.c.i,
if c h a r ( k ) > O.
4
In the first part of this paper we prove two parallel which give evidence characteristic a s.t.c.i,
of the d i f f e r e n t
on Fland on F 4 (Proposition
which were introduced
by
bases
closure of an affine
is positive,
if char(k)
(G-bases)
is zero,
then C 4
1.6). To get our
of ideals
in polynomial studied in
the equations
defining
(Prop. l.4 and [8] ).
is an element of the family of the so called monomial
Now C
to the
then C 4 is
in [ 3] and further
for computing scheme
statements,
of C 4 with respect
(Proposition
Buchberger
, and which yield an easy method
the projective
1.5);
on FI, on F 2 and on F 4
proofs we use the so called Grobner rings,
behaviour
of the field k. Namely, if char(k)
is not a s.t.c.i,
[4]
curve
curves
3 in ~P ,
4 but it is in the "bad" p a r ~ of the family,
in the sense that its projective
392
coordinate ring is not Cohen-Macaulay.
The second section of the paper is
devoted to show that all the monomial curves in ~ coordinate ring is Cohen-Macaulay,
are s.t.c.i,
3
, such that their projective
in every characteristic
(Corollary 2.3). To get the proof we need a good description of the defining ideal of these curves; this was done in [ I ]
(see also [ 2 ] ) and we point
out that it can be easily achieved again by means of G-bases. What we get is that all the members of this "good" part of the family share a special determinantal
structure which allows us to prove our claim by studying some
particular m a t r i c e s ( T h e o r e m 2.2).
i. Let A be the set of the monomials in the polynomial ring k[Xl'''''Xn ]" We may order the elements of A by their degrees and, if they are of the same degree, by the lexicographic order. If f is a polynomial in k[X 1 .... ,Xn ] we denote by M(f) the m a x i m u m monomial of f and,if I is an ideal of the ring k[X I, .... X ], by M(I) the homogeneous ideal generated by the maximum n monomials of the elements of I. This enables us to give the following definition. Definition i.I. Let I be an ideal in k[X 1 .... ,Xn]. A set of elements {fl,...,fr} C I is called a G-base of I __ifM(I)=(M(fl),...,M(fr ))" Lemma 1.2. If {fl ..... f } --
is a G-base of I, then I=(f .... ,f ).
r
Proof. Assume by contradiction that such that M(g) < R ( f )
1
r
(fl,...,fr)C I and let f6 I , f ~ (fl,...,fr)
does not hold for all elements g with this property.
We get M(f) 6 M ( I ) = ( M ( f l ) , .... M(f )) hence for some monomial a and some i we r have M(f)=aM(f
i
a contradiction.
). This implies M(f-af ) < M ( f ) 1
thus f-af
i
6 (f .... ,f ), 1 r
393
For more details on G-bases see
[ 3 ] and
[4 ] h
Now we recall that if f is a polynomial in k[Xl,...,X n] then
f will denote X ] (here 8f n
. .O) .in . k[X . O, the homogeneous polynomial X ~ff(xl/x 0 . .. ,Xn/X is the degree of f); also for any ideal I in k[XI,...,X n] , the homogeneous ideal generated by the forms seen that the homogeneous elements in n~O
h
h
h
I will denote
f with f 6 I. It is easily
I are the forms
h
fX
n with f 6 I and 0
(see [ ii]p.180). Also if I(V) is the defining ideal of an affine
n algebraic variety V in ~% , then the defining ideal of its projective closure inIP
n
is
h
I(V). Finally if I is an ideal in k[Xl,...,X n] , M(I)
e
will
denote the ideal M(I)k[Xo,...,X n] h Lemma 1.3. a) I f f 6 k [ X 1 ..... X n] , then M ( f ) = M ( f ) . e h b) __If I is an ideal in k[X I ..... Xn ] then M(I) = M ( I ) . Proof. The first assertion is clear. As to the second one, we have seen h that the homogeneous elements in
h I are the forms
n h n fXo; now we have M( fXo)=
h n n hi) E e =M( f)X =M(f)X and this proves that M( M(I) , while the other inclusion O O is a consequence of a). h Proposition 1.4 • If {fl,...,f } is a G-base of the ideal I, then {hf I , . . . , --
r
is a G-base of the ideal
h
r
I.
r r h e Proof. We have M ( I ) = M ( I ) = ~I M(fi)k[Xo ..... Xn] = ~1 M(hfi)k[Xo ..... Xn] = h = ( M ( f l ) , .... M(hf )). r Let C4 b e t h e q u a r t i c 3
c u r v e i n ]P3k g i v e n p a r a m e t r i c a l l y
by Xo=U4, Xl=u3t,
4
X2=ut , X3=t . Take the standard affine open set XO~0 and put X=Xl/Xo,
Y=X2/X 0 and Z=X3/Xo; t h e n C4 i s t h e p r o j e c t i v e
f
closure of the affine
curve
394
3 4 3 C: X=t, Y=t , Z=t , whose defining ideal I is generated by f =Y-X , I f =Z-XY. Standard techniques in computing G-bases, give a fairly easy way 2 to get a G-base of I, hence ideal of C 4 in 3 .
(Prop.1.4) to get generators of the defining
h If we denote by P this ideal, we know that P= I and we h
get the well known fact that
2
3
P=(F1,F2,F3,F4) , where FI= fl=XoX2-X1,
h
2
3
F2= f2=XoX3-XlX2, F3=XoX2-X2X2 1 3' F4=XlX3-X2" Another well known fact (very easy to be checked) is that C 4 is schematically defined by FI, F2, F 4. Now it is clear that if we can find two polync~ials f and g such that h h h I=rad(f,g) and {f,g} is a G-base of (f,g), then P= I= rad(f,g)=rad( (f,g))= =rad(hf,hg), where the last equality follows from Proposition 1.4 and the third one from [II] p.180
; of course this would imply that C
is a 4
s.t.c.i.. It is well known that C 4 is a s.t.c.i, if c h a r ( k ) = p > O and indeed in this situation we can find f and g with the above described properties. Proposition 1.5. C 4 is a s.t.c.i, on F 1 o___~n and F 4 i_ffc h a r ( k ) = p > O . Proof. We observe that FI, F 4 play an interchangeable role with respect to the standard affine open sets Xo# 0, X3# 0. So let us see what happens on F I. 3 3 3 3 3 4 3 4 Case a): p=3. Then f2 = Z -X Y =Z -Y mod(f I) . Hence g=Z -Y ~ I
and I=rad(fl,g).
3 4 Moreover M(fl)=X , M(g)=Y are coprime, so {fl,g} is a G-base of (fl,g) and we are done. 3p p 3 p p 3 3p 2p Case b) : p~3. We have f2 =(f2 ) =(Z -X ~ ) =Z -3Z ~ + 3 Z
p 2p 2p 3p 3p X Y -X Y .
Now since p~3 we can write 2p=3q+r with l ~ r ~ 2
It follows that
and q > r .
f~P=g mod(f I) where g = z 3 P - 3 ~ z 2 P + 3 X r y 2 p + q z p - y 4 p ; +r(3/2)=r(5/2)+q(ll/2) ~2r+6q=4p,
but r+q+3p=r+q+q(9/2)+
hence M(g)=Y 4p and we conclude as before.
395
However the situation turns out to be completely different if char(k)= 0. Proposition
1.6. If char(k)= 0, then C --
is not a s.t.c.i, on anyone of the 4
"
three surfaces FI, F 2 an___ddF 4 which define it schematically, Proof. Certainly C 4 cannot be a s.t.c.i, on F 2 (even in positive characteristic), since F
is a non singular quadric and C 2
is of type
(3,1) on it. As before
4
we may restrict our attention to F . Assume that there exists a polynomial i G such that P=rad(FI,G);
then G is not divisible by X
and so, if we denote 0
by g the unique polynomial in k[X,Y,Z] Now if X divides M(g), then
such that
h
g=G, we get I=rad(fl,g).
(FI,G) Ci(Xo,Xl) but this is in contradiction
with P=rad(Fi,G) ; therefore X does not divide M(g).Let A be a domain and f, g elements of A such that f~ 0 is prime and c unit in A and n a positive integer.
(f)=rad(g); then g=cf
n
with
So, if A denotes the ring k[X,Y,Z]/(f I) n
which is isomorphic to k[X,Z],we deduce that g=cf2+af I with c 6 k ~ and of n
course we may assume g=f2+afl. To get our claim, show that for every a 6 k [ X , Y , Z ]
it is then sufficient to
and for every positive integer n, X divides
n
M(f2+af I) . Now it is easy to check that this is equivalent to saying that, for every positive integer n,X divides M(h), where h is the remainder n 3 obtained in the division of f2 by X -Y in k[Y,Z][X]. This can be achieved by an elementary computation.
For other remarks on this subject see [12 ].
2. Let C be a monomial space curve in • XI=U n tm-n ,X 2 m >p);
=uPtm-P,x3=tm
3
m given p a r a m e t r i c a l l y by X =u , O
(m,n,p are positive integer such that m > n
and
it is clear that C is the projective closure of the affine space
curve with parametric equations X=t m-n, Y=t m-p, Z=t m. The defining prime
396
ideal of these affine curves has been extensively studied by Herzog in[ 7] (see also
[ 9]); using his results and Proposition 1.4, one can easily
give a complete description of the equations defining the projective q~urve C. We collect these facts in the following theorem which has been p r e v i o u s l y proved by different methods in [ 2 ] and whose proof is left to the reader. Theorem 2.1. The curve C is arithmeticall~.C0hen-Macau!ay if and o n l ~ i f the minimal number of generators of the defining ideal I(C) is less or equal to three. Further if C is arithmetically Cohen-Macaulay, then I(C) is generated by the 2x2 minors of the matrix
X
z b cd[l X2
Ix
x X O 3
x
for suitable non negative inte@ers a,b,c,d,e,f,g,h.
Now if C is arithmetically Cohen-Macaulay we can prove that C is a s.t.c.i. as a consequence of the following more general result (see also[10]). Theorem 2.2. Let R be a commutative ring with identity; let m and n be non negative integers and let I be the ideal generated by the maximal minors of
the matrix M=
IIynx
:II
with entries in R. Then there exist elements f
s
and g in I such that rad(I)=rad(f,g). Proof. We may assume that m~>n; let m=nq+r where 0 ~ < r < n, q > i and let m n n+m fl=y t-sz, f2=y z-xt, f3=xs-y . In the following,if a and b are elements p-I of R, we use the identity (a-b)P=aP-b(k~oak(a-b) p-k-l) which can be easily ~q+l , n _,q+l n(q+l) q+l checked. We have =o =~y z-xn) =y z -xte,where for suitable c in R
397
q we let e=k~ynkzkfq-k=ynqzq+f^(yn(q-l)zq-l+cf2). =u x z
Next we denote yn(q-l)zq-l+
s
y
te
yr
x
z q+l
+cf 2 by a, consider the matrix N=
and call J the ideal r
generated by the 2x2 minors of N. Then J is generated by fq+I • g=y ~e-sz 2
q+l
and f ; now we have g=y r te+z q (ym t-sz)-y m z q t=y r t (e_ynqzq)+zqfl=yrtaf2+zqfl=
3
y =det
m
y
s
zq
r y ta
sqg=det
z which implies J C I. On the other hand
sq x
ym
qn s y
s
sqz q
y ta
sqx_sq-lym+n
iJl =det
r
y
=det
II:
0
s
t
s q z q _ s q - l y n + r ta
y r ta
0
z
m
s q z q _ s q - 1 y n+r at
ym
s
t
y r at
0
rood (f3) = fl (sqzqesq-lyn+rat)
and sqzq-sq-lyn+rat=sqzq-sq-lyn+rtyn(q-l)zq-i
But for suitable d in R we have
mod(f2)=s
rood(f3)
q-I q-I z (sz-ymt) mod(f2).
(_fl)q-l=(sz_ymt)q-I =s q-I z q-i -y m d, hence
q ¥ q+l 2 m n m s g=-fl +fly d mod(f2,f3); now y f +sf +tf = 0, hence y f 6 (f2,f3) 1
2
3
we get fl q + 1 6 (g,f2,f3) " This implies that rad(I)=rad(J)
and
i
Thus,if r=0, J
is clearly generated by two elements and the theorem follows; if r > 0, we apply the same argument to the matrix N and eventually• after a finite number of steps, we conclude the proof of the theorem. 3 Corollary 2.3. Let C be a monomial space curve in ]P which is arithmetically Cohen-Macaulay;
then C is a s.t.c.i.
398
Remark.
It is interesting to compare Theorem 2.2 with the 1
2
that the Segre embedding of IP × ]P
by the 2x2 minors of the matrix
in ~
5
(see
, whose d e f i n i n g ideal is generated
IIxo xl X3
well known result
X4
, is not a s.t.c.i. X
[5 ] ).
References
[ i] Bresinsky H. and Renschuch B., Basisbestimmung Veronesescher ideale mit allgemeiner Nullstelle
Projektions-
(tmo,tom-r tl,t r m-s s m Math. Nach., ° tl,tl)
(to appear). [2] Bresinshy H., Schenzel P. and Vogel W., On liason, arithmetical Buchsbaum curves and monomial curves i n ~
3
, preprint.
[3] Buchberger B., Ein algorithmisches Kriterium fur die Losbarkeit eines algebraischen Gleichungssystems,
Aequa.Math.
4 (1970), 374-383.
[4] Buchberger B., A criterion for detecting unnecessary reductions in the construction of Grobner bases, Proc. EUROSAM 79, Lect. No. Comp. Sc. 72 (1979), 3-21. [5] Hartshorne R., Cohomological dimension of algebraic varieties, 88
(1962), 403-450.
[ 6] Hartshorne R., Complete intersections I01
Ann. Math.
in characteristic p > 0 ,
Am.J.Math.
(1979), 380-383.
[7 ] Herzog J., Generators and relations of abelian semigroups and semigroup rings, Man.Math.
3 (1970),
175-193.
399
[ 8 ]Moller H.M. and Mora F., Grobner bases and explicit free resolutions of polynomial modules,
in preparation.
[ 9 ] Robbiano L. and Valla G., On the equations defining tangent cones, Math.proc.Camb. Phil. Soc. 88 (1980),281-297. [ i0] Valla G., On determinantal intersections,
Comp.Math.42
ideals which are set-theoretic complete (1981), 3-11.
[ ii] Zariski O. and Samuel P. : Commutative Algebra,v. II, Van Nostrand, Princeton,
[ 12]
1960.
Craighero P.C., Una osservazione sulla curva di Cremona di p3, C :{~B3,~3~,14,~4 }, Rend. Sem. Mat. Univer. Padova, 65 (1981), 177-190.
CONSTRUCTING
ENRIQUES
SURFACES FROM QUINTICS IN P-. K Ezio Stagnaro
Introduction,. Let k be an algebrically Enrlques
closed field of characteristic
surface is a non-singular
pg = O, the irregularity 2K is equivalent
projective
zero. By definition,
an
surface with the geometric genus
q = O, the bigenus Po.. = I and the double canonical divisor
to zero in a minimal model of the surface.
We shall construct Enriques
3 starting from quintic surfaces F 5 in Pk"
surfaces
Our surfaces F
will have four tacnodes at the vertices of a tetrahedron, such 5 that there exist two planes 71, ~2 which are both tangent tacnodal planes to F 5 at two vertices
of the tetrahedron.
that the geometric genus p
g
If F ÷ F 5 is a desingularizatlon
of F is zero, because
the planes passing through the four vertices adjoint surfaces
of F5, we have
the adjoint surfaces
of the tetrahedron.
to F
are 5 The bicanonical
to F 5 are the quadrics passing through the four vertices
that the tangent planes to the quadrics
and such
coincide with the tacnodal planes of F : 5
Such a quadric exists and moreover we have that it is unique:
our quadric is
gi-
ven by the two planes ~I' ~2" Therefore F has the bigenus P2 = I. Moreover we construct again Enriques
surfaces
also when two tacnodes are triple
points on F . 5 For all these surfaces F, we have the trigenus P3 = O, namely a tricanonical joint to F 5 must have at the vertices
of the tetrahedron
ad-
a double biplanar point h a
vlng one of the two singular tangent planes coincident with the tacnodal plane to F 5. Such adjoints
do not exist;therefore
F has P3 = O. This fact, P3 = O, is equivalent
to the fact that on a surfaces with pg = O, P2 = i without exeptional kind
(minimal model)
the bicanonieal
divisor 2K is equivalent
to zero.
223-224 or p. 246). Finally we shall prove that the irregularity Hence our surfaces F are Enriques Moreover
curve of first (cfr. [E] pp.
of F is q = O.
surfaces.
for the plurigenera we have P2i = I, P2i+l = O, i > O (lot. cit.). For fu~
ther discussion
see also Ch. VII, § I, pp. 237-247 of the same Enriques'book.
We remark that our surfaces have lines passing through the tacnodes, near system of plane section through a tacnodes against the assumption
contains
in the theorem of Enriques
lines belong to the complete intersection
C
(cfr.
then the li-
a pencil of reducible
curves,
[E] , pag. 72). But such
of Fq with the bieanonical
401
adjoint
;since C is an exeptional
line of first kind, we may consider a surface
without curves of first kind, so the above lines vanisch. the double plane, given by Enriques, two exceptional
ques - Castelnuovo:
surface,
there are
of F is q = O. We use the criterion of Enri-
"The adjoint surfaces
canonical
series"
(cfr.
it may have either no singularities, canonical
as model of an Enriques
lines of first kind).
Now we pro~e that the irregularity
(complete)
(Remember that also in
of degree 2 cut on plane sections
[E] , pag. 118). Now if C is a plane section, or singular points;
series is p - i = 5, or < 5 respectively
nus of C). The adjointquadricsmust
the
so the dimension of its
(where p is the geometrical
pass simply through the four tacnodes,
g~
so they
are a linear system of dimention 5 ~ 5. Q.E.D. Therefore
if F
+
F 5 is a desingularization
of F5, for F we have:
pg = O, q = O, P2 = i, P3 = O. The surface F is an Enriques
surface,
as required.
1.1f we considerthe two planes:
Q: the corresponding
(X 1 + X 3)
quintic F 5 in
(X2 + X 4)
p3 is k
3 2 F 5 : X 1 (X2 + X 4) + 3 2 +X 2 (X 1 + X 3) + 3 2 +X 3 (X2 + X 4) + 3 2 +X 4 (X 1 + X 3) + +(a221oXIX2X3+a22oIXIX2X4+a2oI2XIX3X4+ao221X2X3X4)(XI+X3)(X2+X4 22 22 + a212oXIX3(X2+X4)+aI202X2X4(XI+X3); Now we prove that F the other tacnodes the property
has a tacnode in A
5 (O,I,O,O),
(O,O,l,O),
)+
a2120~ O, a1202# O.
= (i,O,0,0),
the same argument holds for 1 (0,O,O,I). A tacnode is caracterized by
that the tacnodal plane cuts F 5 in a curve which has a singular 4-fold
point at the tacnode
(cfr.
[E] , pp. 84-85).
The tacnodal plane at A I on F 5 is gi-
ven by X 2 + X 4. Cutting F 5 with such plane we get: 4 F5 = O i aI202X4(XI + X3) = O X 2 + X4 =
O
I X2 + X4
=
O
402
which is a curve with a 4-fold point in A
(and also in A = (O,0,i,O)). 1 3 We remark that X 1 + X 3 is the tacnodal plane of F 5 at (0,I,0,O) and at (0,O,O,I)~ X 2 • X 4 is the tacnodal plane of F 5 at (I,0,O,0)
and at (0,0,I,0)
(see the intro
duction). If F is a desingularization
of F5, then for F
we have: pg = O,P 2 = i
adjoint to F ). 5 In this case it is easy to calculate the complete intersection
P3 = O.
(Q is the unique bicanonical
the theory, we know that it is an exceptional
of Q and F5;from
curve of the first kind.
3 in P 5 k having two triple points and two tacnodal points such that the two tacnodal planes 2. It is possible
to construct Enriques
pass both through the two triple points.
surfaces
from quintic surfaces F
The bicanonical
adjoint Q'
(cfr.
[E] ,
pag. 74) is given by the two tacnodal planes which have a double point at the two triple point on F' : 5 Q': x3x 4 3 2 F': blX3X 4
+
3 2 +
b2X4X 3 +
+
a2210X1X2X3
+
al211X1X2X3X 4 + all21X1X2X3X4 + alo22X1X3X4 + aO221X2X3X4 + aO212X2X3X4 +
+
aO122X2X3X 4 + alll2X1X2X3X 4 ~
22
22 2 22 2 2 + a220IXIX2X 4 + a2111XIX2X3X 4 + a2021XIX3X 4 + a2oI2XIX3X 4 + 2
2 2 2
22
22
2
2
b I ¢ O, b 2 ~ O, a2210 ~ O, a2201 ~ O, a2021 # O, a2012 ~ O, ao221 # O,
a0212 ~ O. Again a desingularization
of F 5 has pg = O, q = O, P2 = i, P3 = O.
Remark. If we apply the standard transformation x
1
=XXX 234
x 2 = XIX3X 4 x 3 = XIX2X 4 x 4 = XIX2X 3 to the surface F'5 ' the proper transform of F' is 5
2
403 3 2 o* (F') : a x x 5 2201 3 4 3•2 + a2210x4x 3
+ +
22
22
22
2
2
22
22
+ b x x x +b x x x +a x x x +a x x x +a x x x +a x x x + 2 1 2 3 1 1 2 4 2111 2 3 4 2021 2 3 4 2012 2 3 4 1211 1 3 4 2 2 2 2 2 2 2 +
a
X
x
X
x
1121 1 2 3 4 2
+a
x
x
x
x
+a
1112 1 2 3 4
x
x
x
x
1022 1 2 3 4
+a
x
x
x
0221 1 3 4
+a
x
x
x
+
0212 1 3 4
+ ao122XlX2X3X 4 The t r a n s f o r m a t i o n o has changed the coefficients in the follow way: b I ~-+ a2201' b 2 + - + a2210' a2111 +-+ a0122 , a2021 +-+ a0212 , a2012 *-+ a0221 , all21 +-+ all12
, a1211 +-+ ai022
a2111 = a0122,
...
i
F5
is fixed for o
.
So if w e choose b I = a2201, b ° ~ = a2210,
in p a r t i c u l a r c h o o s i n g all the coefficient = i, w e see that 3322 (up to the e x e p t i o n a l planes X l X 2 X 3 X 4 ).
Bibliography [E]
F. Enriques, Le superficie algebriche,
Istituto Belzoni
di M a t e m a t i c a
Applicata
7 - 35131 P a d o v a
(Italy).
- Facoltd
Zanichelli,
di I n g e g n e r i a
Bologna
(1949).
- Universit¢
- Via
PRYM SURFACES AND A SIEGEL MODULAR THREEFOLD by Gerard van der Geer
The topic of the talk is the modular variety
r2(2)\H 2 , where
H2
is the Siegel upper half space of degree
ker{Sp(4,~)~Sp(4,~/2~)} complex manifold. pactification) copies of
. The quotient
to a singular algebraic variety
F(2)\H
F(2)\H*
and
15
Proposition. The
(resp.
is a non-compact
F2(2)\H ~
F(2)\H .
In this way we add
l-dimensional (resp.
"
projective
Sp(4, ~ 2 )
(153,153).
Sp(4, ~ 2 )
If we number these six sets then each
There are
l-dimensional boun-
on them defines an iso-
with the syulnetric group
may be denoted by
l-dimensional
4-dimensional symplectic vector
The configuration is of type
dary components and the action of
component
15
O-dim.) boundary components of
six (maximal) sets of five disjoint (compactified)
morphism of
by adding 15
r2(2)\H ~ .
2-dim.) linear subspaces of a ~2
(Satake com-
points such that each of these points occurs
correspond bijectively to the totally isotropic
space over
r2(2) =
forming a configuration which is explained by the modular
interpretation of
r2(2)\H ~
and
It can be compactified in a minimal way
as a cusp of three copies of lines
P2(2)\H2
2
S6 .
l-dimensional boundary
%ij ' l~i<j~6 , after the two of these sets
405 to which it belongs• The equations
+ 832 + yT 3 + 6(T 2 -31T3) +
~,...,e
wi th
£ ~, B2-4ey-4~E = ]
e
= 0
32) 32 T 3 31
3 =
define a surface
F|
in
F2(2)\H ~
consisting of ten components, each of which is isomorphic to
(r (2) \H*) 2 .
Consider the usual theta functions
•
m' 3t
e (3,~) = ~ exp(2~1{½(~+~ ) m ~2Z 2
with
m = m'm"
sixteen
ten are even as a function of
el(3,0)
are called even. 8~(~,0)
Each
of weight el(3,0)
2
on
t
M2
~
F2(2).
and
Fm(m)\H~
~
2
on
and the image is a quartic threefold
t 42 _ 4t 8 = 0 t8 = ~ el6m
m
F] . The
5-dimensional
2 .
F2(2)
define an embedding
~4~9
X
Of the
The corresponding
T mod F2(2) ~ (...,e~(3,0),...)
t4 = ~ e8m and
~ E E2.
and these yield
They span the
of modular forms of weight
Theorem. The modular forms of weight
where
m"
(~+~)})
vanishes on one component of
are not linearly independent.
vector space
m'
+ (~+~)
a fourtuple of zeroes and ones,3 ~ H 2
em(T,~)
modular forms
m'
(~+~)
given by
406
The singular locus of the "boundary". Riemann's
p
tangent space
consists of the fifteen lines coming from 2 t 4 - 4t 8 = 0
The relation
theta formula.
Theorem. Let
X
We use geometric methods
be a non-singular T
of
was found by Igusa using
X
at
p
point of
with
X
to derive it.
X . The intersection
is a Kummer surface if
of the p does
P not lie on the surface (with level
If
p
2
F l . In this case the moduli point of
structure
induced by the fifteen singular
is a non-singular
point of
a quadric surface with multiplicity which
p
belongs.
curves
(~
l
X
lying on
T
then
T NX P
is
Fl
to
It carries the structure of a product of two Ku~ner
with four points marked).
~m(~,~) = @~(T,0)@~(~,~) L
lines) equals p .
2 , namely the component of
In order to prove the theorem one considers
bundle
FI
T NX P
on
A
T
,
for
the
m
even.
Abelian
satisfy the linear relations
surface
the ten functions
They define sections of a line corresponding
satisfied by the
04(T,0) m
to
• . Since
they
we have a linear
map M2
~
H°(A,LT)
@4(T,O) -~ o4(T,-)
and a corresponding
projective embedding ~(H°(AT,L
One verifies
that the
c
m
(
V
:linear dual).
satisfy
(~ c~)2 This implies that the image of Kummer surface,
) v) ~ ~ (M2 v)
is contained in
_ 41 1 6 m A
T
in via
= 0
.
(H°(A~ 'LT)v) , which is a ~ . Since
H°(AT,L )_ is
4-dimens-
407
ional, this Kummer surface is cut out on image
p
of the origin of
AT
must be the tangent plane of
X
by a hyperplane.
must be a singular point, X
Let ture
C
threefold
be a non-singular
(i.e. the ramification
Denote by and set any
j(k)
in
~
points
j(1)
Since points
Xl,X 2
on
order
2
on
to
this yields an isomorphism 0c®L =
J .)
If
of
L-t
0C-mOdules
if
Xl#X 2.
with the involution of
C ,~ ~ C face
~
C
such that
C
xI
determines
and a
Spec(0ceL
2~#2x
2-dimensional :
carries a level
2
on
C ,
Moreover,
are the points of C
corresponding which turns the
0c-algebras.
Let
)
j(2)
for all
are unique and
P , called the Prym variety
P
k
Xl,X 2.
(y~4q-y)
induced by the hyperelliptic x
on
It is
is compatible involution
C.
i
This implies
x1#x 2 . Then the covering
principally polarized Abelian surP = ker {Nm: J a c ( ~ , # )
over, the situation also yields a symplectic hence
J2
with ramification points
x2
struc-
q] (=q),q2,...,q6).
~: L ®2 ~ 0c(-xl-x2)
The involution on $2C
2
2a = (x]+x2).
sheaf on
into a sheaf of
C . We now shall assume that
that for this
with a level
is surjective, we can find for
is the invertible
he the double covering of
of
is contained
are marked
is unique if e ~ q + J2 " (Here
~ ,~ = non-singular
2
C ~ • I
$2C ~ j(2)
Xl,X 2
sheaf
p
this requires an explanation.
curve of genus
the pair
~
Of course,
to
the variety of divisor classes of degree
j = j(0).
~
the hyperplane
at this point.
We thus see that the Kummer surface belonging in the Siegel modular
Since the
structure.
isomorphism
In this way
~,~
~ J} . More~: J2 ~ P2 ' ~ C
deter-
408 mines a point of dent of
F2(2)\H 2 , the moduli point of
P . (This is indepen-
~ .) !
If
2~ffi2q then there is a whole
~
of possibilities x E C/i --~ ~,1
: L ®2 =, Oc(-X-i(x))
This forces us to blow up yields a non-sigular then
~
~,~
J (I)
a multiplicative
~(I)
surface
is singular and
P
variety associated
~ E ~(I)
points. just
the sixteen curves
with
this with p
The essential theta functions as given by
and
~ ~ C/icj(1)
to it induces an embedding
Combining
TpNX
2~=2x
to the moduli point of the Prym
~
F2(2)\H ~ ,
is the Kummer surface obtained from
contracting
if
This
is an extension of an elliptic curve with
KC KC
Secondly•
q + J2"
group.
Theorem. The map which sends
where
in the sixteen points of
~(1)/i
(i: y ~ 2q-y) by
n + C/i , ~ E J2 ' to ordinary double F2(2)\H ~
the moduli point of
~4 C
we find that the image is (or
K C) .
ingredient for the proof is the relation between the
(denoted by
~ )
on
P
and the theta functions on
Fay : n4(0) = c(~,~)@2(~,O)O2(T,~)
for some non-zero constant to prove our second theorem.
c(~,~).
Note that this relation can be used
409
To make the picture complete one has to consider degenerations. do this one has to look at a certain non-singular model of by blowing up forms.
X
along the 15 singular
We have the following
X
To
obtained
lines by means of the ideal of cusp
cases.
a)
an elliptic curve with an ordinary double point.
b)
two elliptic curves intersecting
transversally
in one point.
One or
both of them may also be a rational curve with one ordinary double point provided
that the intersection point is non-singular.
c)
a rational curve with two ordinary double points.
d)
two
IP l's
intersecting
transversally
We consider double coverings
i)
is non-singular
double points of ii) ~
reg
~ C
Let
reg
Jac(~)
(resp. C).
Then
such that
C, is ramified in two points.
(resp. Jac(C)) P = ker { N m
a Pl~cker surface,
be the generalized Jacobian of
: Jac(~) ~ Jac(C) } is again a principally
The corresponding
Kummer surfaces are :
i.e. a quartic surface with one double line and
eight isolated double points; plane containing
~-~ C
except for ordinary double points lying over the
polarized Abelian surface. a)
in three points.
it is the intersection
exactly one of the singular lines o f
of
X
with a hyper-
X .
b)
a quadric with multiplicity
F!
and the zero divisor of a
e)
a quartic surface with two double lines and four isolated double
points;
it is the intersection
two; such a quadric is a component of e 4 (r,0). m
of
X
with a hyperplane passing through
410
exactly two singular
lines.
d)
a quartic surface with three double lines; it is the intersection
of
X
with a hyperplane
containing
exactly three of the singular lines.
There are fifteen such surfaces and they are given by the equations ~TI
+
BT2
+ YT3
+
2 ~(~2-~IT3 ) +
e = 0
2 with of
B - 4ay - 4~E = 4. HxH
Each of them is a suitably compactified
quotient
by the group : al ~ a 2 (rood 2)}
{(al,~2) c S L ( 2 , = ) 2 and parametrizes curves.
Abelian surfaces isogenous
If we change the action of
S6
to a product of elliptic
by an outer automorphism
then
X can be defined by (~ x 2 2 i ) -4[
x i = 0, in
~5
with the action of
six coordinates.
S6
x4 i" = 0
being given by the permutation
of the
Our fifteen surfaces are now given by .U. (xi-x~)a = 0. l<j
For a curve of genus greater than
2
we get a rational map
! $2C ~ A 2 g,2 ' where
l2
2 S C = { ac and
A
g,2
j(1)
: 2~ES2C c
j(2) }
is the level two moduli variety of principally polarized
Abelian varieties double cover
of dimension g . It is defined by associating
C~C
its Prymvariety.
with two ramification Since the surface
the Schottky problem
$2C
to a
points the moduli point of is intimately connected
(cf. Curves and their Jacobians by Mumford,
to
lect. 4)
we hope that our study of the higher genus case will contribute to a solution of the Schottky problem.
411
REFERENCE S
-- Fay, J.D.: Theta functions on Riemann surfaces. Lecture Notes in Math. 352. Berlin, Heidelberg, New York : Springer Verlag 1973. -- van der Geer, G.: On the geometry of a Siegel modular threefold. Math. Annalen
260, 317-350 (1982).
-- Hudson, R.: Kummer's quartic surface. Cambridge : Cambridge University Press 1905. -- Igusa, J.: On Siegel modular forms of genus two (II). Am. Journal of Math. 86, 392-412 (1964). -- Mumford, D.: Prym varieties I. In: Contributions to analysis, pp 325350.
London, New York : Academic Press 1974.
-- Mumford, D.: Curves and Their Jacobians. Ann Arbor : The University of Michigan Press 1975.
