1 Preliminaries
In this
chapter of preliminaries
we
review from
[3], [6], [17]
and
[31]
some con-
cepts and well-known results about birational maps of surfaces and weighted clusters and derive maps in
chapters
some
applied to plane Cremona proofs of these consequences have been
consequences that will be
2 and 4.
Only
the
included. this
Throughout
irreducible surface
chapter
over
[3]
notations of
with
surfaces,
Ll
Blowing-ups
we
by surface a smooth projective complex numbers C. When dealing adopted.
shall
mean
the field of the are
Definition 1. 1. 1 Let S and S' be two surfaces. A rational map !P : S --+ S' is a morphism from an open subset'U of S to S' which can not be extended to any
points to
open subset. The of 4i. A birational, map P
larger
some
points :
S'--+ S' is
non-empty open subset V of S is
Remark 1.1.2 Since
we
S
in F a
an
-
U
are
called
fundamental
rational map whose restriction
isomorphism.
will deal with rational maps that are birational maps are dominant and their composition
,between irreducible surfaces, these
always makes Let D be
sense.
a
divisor
on a
surface S. The set of all effective divisors
on
S
linearly equivalent to D will be denoted by I D 1. 1 D I can be identified with the projective space associated to the vector space HO (Os (D)),. where Os (D) is the invertible sheaf corresponding to D. As long as no confusion may result, we use
the
same
notation for
a
divisor and its class in Pic S.
subspace C of IDI is called linear system on S. complete if C ID1. The dimension of C is by definition its dimension as a projective space. We say that a generic element of C has a property P if elements in a non-empty Zariski-open set of the projective space C have the property P. We say that C has a curve G as a fixed component if every divisor in C contains G. The fixed part of C is the biggest divisor F that is contained in
Definition 1.1.3 A linear C is said to be
every element of C. Then the linear
system C
M. Alberich-Carramiñana: LNM 1769, pp. 1 - 28, 2002 © Springer-Verlag Berlin Heidelberg 2002
-
F has
no
fixed part.
1 Preliminaries
Let P
:
S
--+
F of !P form
S' be
a
birational map of surfaces. The fundamental points points of S Q3] IIA).
finite set of
a
Definition 1.1.4 Let C be
an irreducible curve on S. Denote by 4i(C) the -P(C F) in S', which will be called the image of C. The direct 0 if -P(C) is a point, or P,, (C) image P,, (C) of C is either !P,, (C) -P(C) if -P(C) is a curve. We define P(D) and !P.(D) for any divisor D on S by
closure of
-
=
=
linearity. Let D be
4i*(D)
a
divisor
on
S'. The pull-back of D by P is denoted by
and is called the total
We shall maps, the
see
transform of
(cf. [31 IIA).
D
that birational maps are composites of elementary birational to which the rest of this section is devoted.
blowing-ups,
Definition 1.1.5 Let S be
a surface, and let p be a point in S. We denote by H : 3 -4 S. The restriction of the morphism H to H-'(S jp}) is an isomorphism onto S fpj, and E := H-1(p) is isomorphic to P1, is called the exceptional divisor of H and can be identified
the
blowing-up of p
on
S
-
-
with the tangent directions For
[6]
3.1
a or
on
S at p.
construction of 1Y and its basic
[31]
Definition 1.1.6 Let H and consider
denoted
by 1
properties
see
for instance
[3] IIJ,
V.3.
S be the
blowing-up
of
a
point
p
on
S,
S. The image of C by the birational map H-1 is and called the strict transform of C (after blowing up p).
a curve
C
on
Definition 1.1.7 Let p, be
a
point
in
a
surface
S,
let
Os,p
be the local
ring
of S at p, and let Mp be the unique maximal ideal of Os,p. Suppose, C is a curve on S, and f E Os,p is a local equation of C at p, then the multiplicity of C at p is the integer ep(C-) for which the relation f E holds. Clearly ep(C) > 0 if and only if p belongs to C. Lemma 1.1.8
S
of form
an
([3] 11.2, [6] 3.2.1)
irreducible
curve
C
on
=
The total
S that has
i
+
A4ep(C) -,A4ep(C)+l
transform after blowing up p on multiplicity ep(C) at p has the
ep(C)E
.
S be the blowingProposition 1.1.9 ([3] 11.3, [31] V.3.2, V.3) Let H up of a point p on a surface S, and let E be the exceptional divisor of 17. 1.
There is
an
isomorphism Pic S E) Z
(D,, n)
Pic'3 1Y
+ nE
.
1.1
2. Let C and D be divisors
S. Then
on
ZT 1
=
C
ZT-E
=
0,
=
_1
-
E 3.
Blowing-ups
Projection formula: let C be
a
2
divisor
-
D
.
S and let D be
on
a
divisor
on
3.
Then
C
ff rs div(w) 2-form on S, then =
is
a
-
D
=
C
-
canonical divisor
(H.D) on
the canonical divisor
r.y
=
Definition 1.1.10 Let 0 be
div a
(11* (w))
point
=
S, 'where w is a meromorphic 3 that corresponds to rS by
on
H* rs + E
in S. The
exceptional divisor
E of blow-
S will be called the first infinitesimal neighbourhood of 0 on S and its points will be called the points in the first infinitesimal neighbourhood
ing
up 0
on
of 0 (on S). If i > 0, we may define by induction the points in the i-th infinitesimal neighbourhood of 0 (on S) as the points in the first infinitesimal neighbourhood of some po'int in the (i 1)-th infinitesimal neighbburhood of 0. In the sequel we will often drop the adjective infinitesimal by saying just neighbourhood instead of infinitesimal neighbourh'ood. The points which are in the i-th neighbourhood of 0, for some i > 0,- are also called points i7ifinitely near to O.'Sometimes the points in S will be called proper points in order to distinguish them from the infinitely near ones, as the word point will be used for both kind of point. Let p, q be two points in S proper or infinitely near. We will say that p precedes q and write p, < q if and only if q is infinitely near to p. We will write p < q if q is equal or infinitely near to p. The relation < is a partial ordering and will be called the natural ordering of the infinitely near points. -
only concerned with the blowing-up of a single point. blowing-up of a subset of proper or infinitely near points in the surface S, which essentially consists of the successive blowingups of all the points, provided that after each blowing-up Sj --+ Sj_1 we identify the points not yet blown up to their corresponding ones on the surface Until
Now
we
now we were
will deal with the
Si Definition 1.1.11 A cluster in
a
surface S is
a
finite set K of proper
or
infinitely near points in S, so that, for each point p E K, K contains all the points preceding (by the natural ordering) p. The proper points of K are called the origins of the cluster. (Notice that these clusters are union of finitely many clusters in the sense of [6] 3.9). A 'subeluster K' of a cluster K in S is a subset of K which is also a cluster in S. By a maximal point in K we shall mean a maximal point in K relative to the natural ordering on K if no other ordering is mentioned.
1 Preliminaxies
4
Definition 1. 1. 12 A is
an
called
a
pair IC
=
(K, #),
map, will be called
arbitrary
a
where K is
weighted
system of virtual multiplicities for (the
K will be called the
a
cluster and A
:
K
--+
Z
cluster. The map [t will be points,of) the cluster K and
underlying cluster of IC. We will usually multiplicity of the point p.
write I-Lp
=
It(p)
and call pp the virtual Let K be
cluster with
0, in a surface S. We. denote origins 01, points in) K by 1TK : SK -4 S. For a detailed construction of HK and its basic properties see for instance [6] 3.5 and 4.3.
the
a
blowing-up of (all
We outline below
.
.
.
,
the
only the
main features.
Definition 1.1.13 An
ordering -- on the points in a cluster K is admissible only if for any pair p, q E K so that p :5 q, we have p : q. That is, an admissible ordering is a refinement of the natural ordering. if and
Fixed
an
admissible total
ordering -
on
K, then HK
is the
composite
of the sequence of blowing-ups of the points in K following this admissible ordering, and SK is the surface obtained from S after these blowing-ups. This construction is
essentially unique (that is, if SK' is the blowing-up of the points in K following another admissible ordering, then there is a unique Sisomorphism from SK onto SK), and hence all the notions related to HK that will be introduced from now on are independent of the admissible ordering used for
defining
them.
The restriction to
Hil(S
-
101,...,0,1)
of the
morphism HK
is
an
.
isomorphism onto S 101,..., 0,}. The origins of points of the birational map ff, ' : S --+ SK -
Definition 1.1.14 Consider
C
K
are
the fundamental
S. The pull-back of C
by IYK is transform of C (after blowing up K). The direct image of C by ITil is denoted by I K (C) and is called the strict transform of C (after blowing up K). denoted
Both
by e
=
ff, (C)
transforms,
OK
and strict transforms of
a curve
on
and is called the total
and
I K,
may be also obtained as the iterated total the blowing-ups composing LIK.
C, respectively, by
Consider the sets
Kp=jqEK:q--
and the proper
blowing-up HKP
point
in
SK,,;
let
:
Hp
SK,, :
X
S of the cluster
SK,
be the
Kp in S. Then p is a blowing-up of p and denote
C X its
exceptional divisor. We identify the points not yet blown up to their corresponding ones on the surface X. Observe that KP' is a cluster in X and let -UK' : SK --+ X be its blowing-up. We have HK 11K,, o Hp o ITKP,. by
Ep'
=
P
1. 1
Blowing-ups
Definition 1.1.15 Consider the divisor EK := 11; ' (101, on the -, 0,,}) surface SK. It is called the exceptional divisor of HK. Each irreducible component of EK is the iterated strict transform of the exceptional divisor of one -
of the
blowing-ups composing HK
-
isomorphic to P'. We write corresponding to p E K, namely
and hence it is
EPK
for the irreducible component of EK
EK
equals
-K' P
UEPI)
Pand
will be called the strict -K
the
p-exceptional component 7-KP' ,
(EPI)
morphism 11K. We write EP for the total transform p-exceptional component of HK.
of
and call it
the total
By induction
on
OK
blowing-ups, easily
follows that
1. Each two
not meet
meet
-K
-K
K, then either EP_ and Eq do holds if and only if p < q. Definition 1.1.17 Let K be on
of
of the irreducible components of EK either do or transversally at a single point, and no three of them have a common point. Given two points p, q E K, p -< q, where -< is an admissible ordering on
Lemma 1.1.16
2.
the number
S. We define the
denoted proper vector
by ep (C),
point
p in
as
SKP.
on
K or
REP_
-=K
D
Eq
,
and the inclusion
at the
infinitely
near
The vector eK
the number
OK
a curve on
(C)
at the
of
=
(ep(C))p(=-K
a curve
point
the mult 'iPlicity of the strict transform
Lemma 1.1.18 Let C be
of C
meet
cluster in the surface S and let C be
multiplicity of C
of effective multiplicities of C
By induction directly inferred:
up K
a
not
p E
1 Kp
K,
at the
will be called the
points of K.
blowing-ups
S. The total
and
applying
1.1.8 it is
transform VK after blowing
is +
E ep (C)-==K EP_ PEK
Clearly I K
can
be obtained from
e
by removing all
its
exceptional
components. Definition 1.1.19 Let p, q be a pair of proper or infinitely near points in S. The point q is said to be proximate to p if and only if p < q and q belongs, as a proper or infinitely near point, to the exceptional divisor of -blowing up p, i.e.
eq(EP')
=
1. It is clear from the definition that q lies either
neighbourhood of p
or on
the first
neighbourhood of
a
on
the first
point proximate
to p.
1 Preliminaxies
The
multiplicities of
a curve
follow the law:
Proposition 1.1.20 (Proximity equalities, [6] 3.5.3) infinitely near) point p on a curve C
ep(C)
1:
=
(proper or
For any
eq (C)
q prox. to p
Definition 1.1.21 A non proper
point
p in S is
proximate
to either
one or
two of its
preceding points, as p lies either on one or two components of the exceptional divisor EK, taking the cluster K as the set of (proper or infinitely near) points of S preceding p. We say that p, is a free point in S if it is proximate to just one point. Otherwise p, is called a satellite point in S. The notions of free and satellite points depend on the surface S: the same p may be satellite if considered as infinitely near point in S and free
point
if considered
blowing
as
infinitely
point in the surfa'ce SK, obtained from S by points preceding p.
near
up the cluster K of
Example 1. 1. 22 The points pi and P2 in the first and second neighbourhood X3 are both respectively of the origin p that lie on the planar cusp C : y 2 proximate to p. The point p, is free and P2 is satellite in the plane. Put =
K
=
fP; Pl) P2 1.
Then ep,,
K "
of
1.1.20),
PP2
There is
proximity
[26] IV.I [6] 3.9.
-
a
=
EI
,
P21
(C) + eP2 (C)
-==K
E
=
P1
=
1+1
_K -K E and + E Pi
P2
=
2
=
-K
EP
ep (C) _K
=
EPI
+
(as an illustration -K EK + EP't I
Pi
2
way of
representing a 'cluster K in a diagram that encodes the points of K. It was introduced by Enriques called Enriques diagram. Let us give its definition following
relations between the
and it is
Definition 1.1.23 Let K be of K is
a
cluster with
a
single origin 0.
The
Enriques
tree-graph where each vertex represents a different point diagram in K and the vertex representing 0 is taken as the origin of the graph; each a
edge joins a pair of vertices that represent points one of which is in the first neighbourhood of the other. We name each vertex as the point it represents. The edges are of two different kinds, straight or curved, according to the following rules: 1. If q is
2.
a free point lying on the first neighbourhood of p, then the edge joining p and q is smooth curved and, if p :A 0, has at p the same tangent as the edge ending at p. Suppose q is a point in the first neighbourhood of p. Then -the edges connecting all points proximate to p in the successive neighbourhoods of q are shaped into a line segment which starts at q and is orthogonal to the edge pq at q.
1.1
Blowing-ups
A cluster with more than one origin is represented as the union of the Enriques diagrams of the subclusters comprising the infinitely near points to each origin. For the sake of clarity, we use the additional convention that the proper points (i.e. the origins of the cluster) are represented by black-filled circles and the infinitely near ones are represented by grey-filled circles.
the
Observe that the free points are, represented Enriques diagram, while the satellite points
on
the smooth curved part of the straight segments.
are on
P7 P5 P6
P2
A P8 P3 P3
A
P4
Enriques diagram with nine vertices, pi is the only origin, points proximate to pi, p8 is satellite proximate to P2) P7 proximate to P2 and the remaining vertices are free.
Fig. axe
1.1.
satellite
Once
an
admissible total
take the cluster K for
K,
we
as a
set of
ordering on a cluster K indexes. If p is a system
will take 1A, written in bold type,
as a
p3 and P4 is satellite
has been
fixed, we will multiplicities indexed by K.
of virtual
column vector
It will be often necessary to index a matrix by a set that includes a cluster K and'an extra element, which will always occupy the first position. For this sake, we will denote by K* the augmented cluster obtained from K by
adding
a new
point
po
preceding all the points
in K and non-related
by
any
proxim.'ity relation to the rest of the points in the cluster K. We will denote by 1P the column vector indexed on K (or on K*, which will be clear from the
context)
p, which
all whose components
equals
are zero
but for the
one
corresponding
to
one.
Definition 1. 1.24 Let p be a point of a weighted cluster IC = IC excess P of the weighted cluster K at the point p is defined as
(K, p).
The
/C
P
=
"P
I-1q
-
' EK q prox. to p
The cluster IC is said to be consistent if all
negative.
excesses
P 1C,
p E K
are non-
1 Preliminaxies
8
R-om 1. 1. 18 and the definition of proximate point -K
1.1.25
Corollary
-K
Ep-
=
EP'
'I
Proposition 1. Let K
=
Jpl,
.
.
.
,
p,}.
Eq
P!IKtop ,
.
Keep the above
1.1.26
obtain the expression
-=K
E
+
we
notations.
Then there is
isomorphism
an
PiC S E) ZO' -24 PiC SK -K
(D, nl,..., n,) 2. Let C and D be divisors
+
on
n,EP-.
-==K
+
+
n,Ep-,,
S. Then
e
=
C
=
0,
=
0
D
-
-K
Ep-K
EP'
for any point p in K. 3. For any couple p, q E K is
1
of different points EPK 1, where rp _rp -K. EK the number of points in K proximate to p, and EPI equals either q K
if EP,
-K Eq' 54 0,
in K that
points
4.
n
i.e.
-
if
of the points p ,or q is maximal among to the other, or equals 0 otherwise.
one
the
proximate
are
For any p, q E K -K
Ep-
O
-1
=K
Eq
=
if p,=
q,
otherwise.
5. For any p, q G K
EP'
6.
-
-1
if p
=
q,
1
if q
is
proximate
O
Eq
Projection formula: let C be
a
to p
otherwise. divisor
on
S and let D be
a
divisor on SK.
Then
CK 7.
If rs 2-form
=
div(w)
on
S,
is
a
-
D
=
C
(I.TK,,D)
-
canonical divisor
on
then the canonical divisor
S, where w is a meromorphic SK that corresponds to rS by
on
11K is 0,
K
r.sK
=
div
(11. w)
=
ff, rzs
+
E Pi
1.1
PROOF: The results follow from 1.1.9
points blowing-ups. As
a
induction
by
Blowing-ups
on
9
the number
OK of.
0
consequence of
1.1.26, using 1.1.18 and 1.1.25,
we
have:
Corollary 1.1.27 1. Let K be a cluster in the plane and curve of degree d that has effective multiplicity ep (C)
let C be
a
plane
at each P E K.
Then d
=
I K )yK -K
eP(C) where H is 2.
The strict written
a
=
e K EP
line.
p-exceptional component EK P
relative to
a
point
p, E K
can
be
as
EP-'
Mq,pEq qEK
where Mq,p
-K
K
:--
-Er'
-
Eq
Definition 1.1.28 The proximity relations between the points in the cluster K may be encoded in a K x K square matrix PK , called proximity matrix of K
(introduced by as'the entry
on
Du Val the
[22];
q-th
Mq,p
row
=
also
if p,
-I
if q is proximate to p
0
otherwise. matrix of
cluster
0
0
0
00
0
0)
0
0
0
00
0
0
1
0
0
00
0
0
-1 1
0
00
0
0
0
1
00
0
0
0
0
0
-1
0
-1
0
0
10
0
0
0
0
0
0 -1 01
0
0
1
0
0
x
a
1
0
-1 -1 0
0
proximity
0
0
0
00
0
0 0
matrix
1
(0
_EP' Eq -
represented by the En-
-1,1]
PK* of the augmented cluster K* is
K* square matrix
PK*
--K
_K
-':
q,
-1
0
Remark 1.1.30 The
=
0
-1 -1
the K*
defined
1
Example 1.1.29 The proximity riques diagram of figure 1.1 is
(1-1
[6]),
by taking Mq,p and p-th column, i.e. by 1.1.26, 5
see
0
PK
1 Preliminaries
10
The
proximity
PK has all the entries
matrix
and all the entries above the
one
I EP' IpEK K
invertible. Thus
is
f
and
E P
IpEK.
1. A divisor
Ep ] e
basis of the Z-module,Z
to
K ...
map relative to the two bases
identity
eZ
EPP.,
f EK IPEK P
Hence it follows:
1.1-31 Let K be
Proposition
the diagonal equal
on
to zero, and therefore it is I
and PK is the matrix of the _K
a
diagonal equal
on
a
cluster in
surface S.
a
SK -==K
D
=
EPEK apEP
,
ap E Z, is
effective if
Pj'a K where
a
=
2. Let C be
(ap) PEK
an
is
=
! 0,
-
on
K.
SK, linearly equivalent
on
_K
ap c Z. Then C
only if
column vector indexed
a
divisor
effective
and
and ap > 0
EpEK apEp'
for
any p E K.
PROOF: Write D into its irreducible components, which
kPK
p-exceptional components I
Let
us
, J
SK, HK.C
on
is
are
the strict
and assertion 1 follows.
PEK
prove assertion 2. Since C is
and effective
_K
EPEK ap Ept
to
linearly equivalent
linearly equivalent
to
HK.
to
K
EPEK apEp'
_K) (EpEK apEp'
=
0
0 and C has support on the exceptional and effective on S. Thus HK.C K = b divisor of ITK, i.e. C EpEK P EP I bP > 0 As the components of a vector =
-
in
a
basis
claim.
are
unique,
we
infer that ap
El
Suppose S
projective plane, and let
is the
PyK, Ep",
_K
-K
are
to these bases is
.
.
,
of
a
PK-1
a
line in S.
1.1.26
-K'...je EP
P,
matrix of the
identity
map
PK-1
=
be the inverse
(aPq)(p,q)EKxK
Suppose
is
an
admissible
has all its entries above the
ordering
Idpi, sK relative
are
of on
the
proximity
K.
diagonal equal to zero, those diagonal are all
to one.and those below the
all
on equal diagonal non-negative. 2. Let p, q E K, p -< q If q is a free point, and hence single point r E K, then
the
By
-
cluster K.
The matrix
H be
and
Ep-',,.
SK and the PK*
Lemma 1.1.32 Let
1.
.
two bases of Pic
matrix
for each p E K, and hence the
bp,
=
-
-
a
q
=
ap"
.
it is
proximate
to
a
1.1
3. Let p, q E
just
two
K, p -< q. If q points r, and r2
is
K, then
apq 4. Let
PROOF: Let
PKPK_1
=
p -< q. Then q is
K,
p, q E
aq
denote the
a" + 12 ap P
infinitely
q-th
row
I and the definition Of PK
=
a
it
q
11
satellite point, and hence it is proximate to
a
in
Blowing-ups
+
q
near
to p
if
and
PK-1.
of the matrix
easily give
only if apq The
> 0.
equality
the relation
1: j ,qar rEK 'r -
where
(1.1)
J,,q
=
1 if q is proximate to r, and 1, 2 and 3 follow.
J,,q
=
0 otherwise. From relation
assertions
Let
prove assertion 4. We
us
to p, then
belongs
apq
to. If q
going
to show that if q is
infinitely near on the order of neighbourhood q proceed by lies on the first neighbourhood of p, then are
> 0. We
induction
aq > 1 > 0 P
according
to assertions
1,
-
2 and 3.
Assume that q lies on the n-th neighbourhood of p. Let r E K be the point whose first neighbourhood q lies on. Then r lies on the (n I)-th neighbourhood of the point p, and by induction hypothesis > 0. Now, -
ap'P
owing
to assertions
1,
2 and
3,
we
obtain
aq> a' > 0, P P -
as
wanted.
Observe that,
according to assertion 1 if apq is not positive, then apq van going to see now that if q is not infinitely near to p, then apq 0. Assume first that q is a proper point. Then from (1.1) we have ishes. We
are
=
a
q
q
and hence
a
Assume
=
0.
that q lies on the n-th neighbourhood of a proper point one of the points which q is proximate to, then r lies the m-th neighbourhood of the point 0, with m < n 1. By induction
0 E K. If on
q P
now
r
E K is
-
hypothesis apr
=
0.
Hence, owing
to assertions 3 and q
and
we are
done.
n
=
0
4,
we
obtain
1 Preliminaxies
12
Example
1.1-33 Let
PK be the proximity matrix appearing in 1.1.29. Then 100000000
110000000 211000000 311100000
P-1 K
110010000 110001000
220010100 321000010
321000011) Definition 1.1.34 The intersection matrix NK of the cluster K is square matrix defined p-th column.
K by taking EP ..j qK t
An easy computation Lemma 1.1.35
NK
=
as
the entry
the
on
q-th
a
K
row
x
K
and
using 1.1.26, 3 gives the relation:
-PtKPK-
Definition 1.1.36 Take C
a curve on
S. Let
us
write
e=ff+E vp (C)-K EP' PEK
Each
vp(C)
is
a
non-negative integer which will be called the effective p-value (O)PEK and we call it the vector of effective values
of C. We put VK (C) = (VP of C at the points of K.
Clearly
from the
the
definition,
proximity
matrix of K relates
multiplici-
ties and values:
Lemma 1.1.37 For any
curve
VK
Definition 1.1.38 If C is in S with
C
(C)
S,
on
=
PK'eK(C)
a curve on
S and Q
cluster K, we say that the the surface SK
underlying
if the divisor
on
=
(K, v)
curve
is
goes
a
weighted
cluster
(virtually) through
K
CQ
VPEP PEK
is effective.
weighted
OQ
is called the virtual
cluster Q.
transform of
the
curve
C relative to the
1.1
If
points
eK(C)
is the vector of effective
in the cluster
K, then. the
Blowing-ups
13
multiplicities of the curve C at OQ can be written in
virtual transform
the the
form
OQ
=
I K
+
_K E UP(C) EP'
PEK
where the vector
UKM
=
(UP(QpEK
UK (C)
Definit on
1.1.39 If
we
PK1 (eK (C)
=
have the
say that the
we
multiplicities equal
curve
C goes
=
-
as
V)
of vectors
equality
eK(C) then
is obtained
V)
through the weighted cluster Q with effective
to the virtual
ones.
Example 1.1.40 Keep the notations of example 1.1.22. Figure 1.2 shows Enriques diagram of the cluster K, in which the (effective) multiplicities of C at the points of K have been indicated. We have eK (C) (2, 1, 1), the
=
A
2
140 P2
1.2.
Fig.
VK(C)
(2,3,6)
Enriques diagram of the CUSPY2
.
X3
at the
origin.
and
PK
1
0
0
-1
1
0
-1 -11
(K, v),
Put Q 4 x
at the
eK (D)
=
with
1.
The
curve
2.
The
following The
=
eK
and consider the
as
=
Lemma 1.1.41
a)
(C),
planar tacnode D y2 _K _bK + Epl, origin. Then D goes virtually through Q andbQ (2, 2, 0) and VK (C) (2, 4, 6). v
With the notations
b) UK(C)
above:
C goes virtually through Q if and only three assertions are equivalent:
curve
virtual
as
C goes
ones. =
0.
through Q
with
if
UK
(C)
> 0.
effective multiplicities equal
to the
1 Preliminaxies
14
c)
The virtual
C
point.
and the strict
a
curve
going
ep(C)
Then
>
(K, v)
through
i
K of
the
curve
and let p E K be
a
proper
vp.
PROOF: Assertions 1 and 2 follow assertion 3.
transform
equal.
are
3. Let C be
transform OQ
directly
from the definitions. Let
us see
1.1-41
By
up(C)
0 <
Since p is proper,
according
=
I'P-1 K (eK(C) P
1.1.32, 11P P-1 K
to
ep(C)
0 <
-
=
-
1', P
V)
(1.2)
-
and
substituting
in
(1.2)
El
vp
.
(K, v) be a weighted cluster in S. If there is Proposition 1.1.42 Let Q with C a curve effective multiplicities equal to the virtual going through Q =
ones, then
Q is consistent.
PROOF: Since the relation ities 1.1.20 that
eK(C)
the effective
satisfy
v holds, using the proximity equalmultiplicities of a curve, we are done. =
F1
PK1V will be called the system
Definition 1.1.43 The vector V
of virtual
A system of virtual multiplicities determines a system of virtual values for the same cluster and conversely. Thus, to define a weighted cluster, once its points are given, it is equivalent values of the
to g,ive either
a
(K, v).
cluster Q
weighted
system of multiplicities v,
or
its
corresponding system
P
K-1 V
of virtual values.
(K, v)
Let
be
weighted
a
cluster in S and let C E S be
a
whose vector of effective values at K is VK (C). Note that in terms of according to 1.1.41, C goes virtually through Q if and only if
VK(C) and C goes if and
only
through Q
with effective
-
V >
curve,
values,
0,
multiplicities equal
to the virtual
ones
if
VK(C) The
1C excess
P
of
a
Lemma 1.1.44
IC
P
=
Next result focuses is
be
weighted
cluster IC
(K, M)
at
a
point
p E K
(1.1.24)
be written in the form
can
a a
cluster in line in
Ip2'
p2.
AtPKIp on
where
the
an
=
WPtK PKIp
case
=
2 projective plane P and ordering has been fixed. Let
that S is the
admissible total
-WNKlpK H
1.1
Lemma 1.1.45 Let C be
SK linearly equivalent
a curve on
=--K
Blowing-ups
15'
to the divisor
-K
aoH
1: apEp
-
PEK
with ap E then D goes
=
Z, and let Ka be the weighted cluster IIK,, (C) is
a
curve
on
and its virtual
virtually through ICa
PROOF: The effective divisor C
D
=
F-pEKypEp'K,
eK(D)
ao >
i4 0,
0),
D
to C.
,
degree
of the
curve
D
on
following
p2 is
column
K
on
aK
=
7K
=
bK where
ao
as
1.1.26. Consider the
by
ao,
If
j5K
with -yp > 0. The
H, which equals C,
-
vectors'indexed
ao (in particular transform br-' is equal
be written
can
C=E + where E
(K, laPIpEK).
p2 of degree
=
(aP)PEK
('YP)pEK PK1 (eK (D)
is the vector of effective
points of the cluster
-
aK)
multiplicities of the
curve
D at the
K. We have
E
=
C
bK
_
E
-==K
(ep (D)
-
ap) EP
pEK
which
implies bK
Hence, according
to
1.1.41,
,DlCa
f)K
=
D goes
^IK > 0
virtually through the weighted
cluster IC a
and +
_K 1: bAt
_K
=D
+E=C.
pGK
The
infinitely
tiplicity
of two
near
Theorem 1.1.46 curves
on
a
points give a geometrical idea of the intersection mula point by means of a formula due to A Noether:
at
curves
(Noether's formula, [6] 4.1.3)
surface S
multiplicity [C D]o
is
points infinitely
to
-
near
and let 0 be
a
proper
point
finite if and only if C 0, and in such a case
[C D]o -
=
1: ep(C)ep(D)
running for p infinitely
near
and D
to 0.
be two
The intersection
and D share
P
the summation
Let C in S.
finitely
many
1 Preliminaries
16
Even if we do not know the effective
but
only
its virtual
intersection
multiplicities multiplicity:
at
a
multiplicities of one of the two curves, cluster, we have a useful bound for the
Proposition 1.1.47 (Virtual Noether's formula, [6] 4.1.3) Consider a (K, v) in a surface S. Assume C is a curve on S and weighted cluster IC 0, are the origins of K. If D is a curve on S going through K, then 01, =
S
E [C D]O, -
E ep(C)vp
!
i=1
.
p(=-K
Example 1.1.48 Consider the cusp C and the tacnode D appearing in example 1.1.40. Using Noether's formula we compute the intersection multiplicity of C and D at the origin p
[C D]p
=
-
while
ep(C)ep(D)
(C)ep, (D)
applying virtual Noether's formula
[C D]p -
and
+ ep,
we
2
>
Ili
see that in this case
+
2
1/ l
+
we
2
I'P2
=
=
4 + 2
6
obtain 4+ 1 + 1
=
6
the bound is reached.
Definition 1.1.49 Let C be
a
linear system
on
S without fixed part. The
of base points of C is a consistent weighted cluster K(C) defined in the following way. Start by taking the proper points 01,. Os E S so that every divisor of C contains them. For Oi, I < i < s, take the virtual multiplicity v(Oi) equal to the minimal multiplicity at Oi of the divisors in C. Fix i E f 1, s}. Then discard from C the divisors with multiplicity at than Oi bigger v(Oi), and call C, -the family of the remaining ones. If these divisors do not share any point in the first neighborhood of Oi, then our subcluster with origin Oi is just Oi with virtual multiplicity V(Oi). Otherwise take all the points that the divisors in C, share in the first neighborhood of Oi, each point p with virtual multiplicity equal to the minimum of the multiplicities at p of the divisors in C1. Again discard the divisors whose multiplicities are not the minimal ones and look for the points the remaining divisors share in the first neighborhoods of the former ones, and so on. This process is repeated for each 1 < i < s. The procedure clearly ends after finitely many steps, as
weighted
cluster
-
.
.
.
,
V(P)2 PEK(C) for C E C.
< C. C
-,
1.2
(Bertini's theorem)
Theorem 1. 1. 50
fixed
out
Let C be
Weighted
clusters
linear system
a
on
17
S with-
part. Then
1. A
generic element of C goes through IC(C) with effective multiplicities equal to the virtual ones and has no singular points outside of IC(C).
IQC)
2.
is consistent.
3. Either C is
composed of
curves
in
a
pencil,
or a
generic element of C
is
irreducible. PROOF: Notice that in the definition of
divisors in
IC(C)
at each
step
we are
discard-
Zariski-closed set,of the
projective space C, hence a generic element of C goes through IC(C) with effective multiplicities equal to the virtual ones and so, according to 1.1.42, IC(C) is consistent. The rest of assertion
ing
[6]
1 is
a
7.2.
Froin'the definition of
system CK theorem
---:
I OK
:
linear systems
on
weighted
C E C I has
([31]
pencil, or a generic element implies assertion 3. M in
a
1.2
Given
a
weighted
cluster IC
admissible total
an
points.on SK
Of
is irreducible and
CK
non-singular. This
=
(K, v)
ordering,
in
S, with K
=
Jpi,
defines
a
(- vpI EP-.
(-UK)* OSK
=
p, I written
the ideal sheaf -K
WIC
ideals
4eIinear
clusters
Weighted
following
cluster of base
fixed part and no base points. By Bertini's JII.10.9.1) either CK is composed of curves
no
zero-dimensional subscherne of
-
-
-
-
-
vp,
P,
)
S, and the stalks of RIC
are
complete
app.4) in the stalks of Os. Conversely, if I is a coherent sheaf of ideals on S defining a zero-dimensional scheme and whose stalks are complete ideals, then there is a weighted cluster IC in S so that I 'RIC (see [6] 8.3.7). A curve on S contains the scheme defined by IC if and only if it goes virtually through IC (1.1-38). If p E S is an origin of the cluster K, then
([51],
v.II
=
the stalk of RIC at p, is
W)c,p
=
If
E
0S,p: vp(f) : Vp}
where vp (f) is the vector of effective values of the germ of curve f = 0 at the subeluster K(p) Iq E K : p < q} C K and Vp is the system of values of the weighted subcluster of IC whose underlying cluster is K(p), otherwise =
RIC&
=
os'p.
Definition 1.2.1 Two
only
if RIC
=
W)C,.
weighted
clusters IC and IV in S
are
equivalent if and
1 Preliminaxies
18
It follows
directly from
Lemma 1.2.2 Let IC
the definitions: and 10
(K, v)
=
S whose values satisfy V
Then
> T.
(K, v')
=
we
have
for
be two
weighted clusters
in
S the inclusion
of
any p E
stalks
RIC,& Definition 1.2.3 Let /C the order
(K, v)
=
UK&
C
be
weighted
a
cluster
on
S. We define:
of singularity of IC
VP(VP
E
-
2
PEK
the virtual codimension of K
c
VP (VP +
(K)
2 pEK
self-intersection of IC
and the
IC. K
2
E
=
pEK
Clearly, IC Given
a
/C'=
-
J(]C)
+
c(IC)
(1-3)
IC in p2 and
weighted cluster
system of all the plane curves of degree HI (]?2, -HK 0 Op2(n)). by fK (n)
n
a positive integer n, the linear going through K will be denoted
=
The number of
presents
to the
independent conditions that aweighted of degree n is defined as
cluster /C in p2
curves
In (n + 3)
-
2
Definition 1.2.4 If
(1.4)
is
an
imposes independent conditions
dim tK (n) :5 c(IC)
(1.4)
.
equality, we say that the weighted cluster IC to the plane curves of degree n. The integer
1 2
n(n
+
3)
7
C(IC)
will be called virtual dimension of t1c (n) and will be'denoted
By (1.4)
we
-vdim and the curves
equality degree
of
by vdim. f1c (n).
have
holds if and n.
tjc(n)
only
<
dim
tjc(n)
if IC imposes
,
independent conditions
to the
1.2
Using 1.1.26 we product of divisors
express the virtual dimension of
=
(K, v)
be
a
vdim t)c (n)
where rS is up K, C
a
=- n
canonical divisor
VP E
EpEK
I
p
be
(K, v)
=
Increasing the value of p, system of virtual values for
IC'
=
obtained
from V by blowing
cluster in
a
units is to take V
n
thus
plane. Then
line in p2.
a
weighted
a
by K,
the intersection
rs)
-
surface S
the
and H is
E N.
new
(C
-
K
7yK
Definition 1.2.6 Let IC n
on
1C 2
as
cluster in the
weighted
=
fr,(n)
19
F' by blowing up K:
in the surface S obtained from
Lemma 1.2.5 Let IC
clusters
Weighted
defining
a new
surface
S, p E K, n1p as a weighted cluster V +
=
(K, v').
Observe that the virtual
multiplicities of the
weighted
new
cluster 1C,
are
vp'=vp+n, I
V
vq' In the
of p
by point of K,
]?2
with
=
vq
if q is
n
-
proximate
otherwise.
we
will
IC1
just =
say IC' is obtained from IC
(K, v')
Assume that there is
and /C2
plane effective multiplicities equal to the .
to p,
we will often say 10 is obtained from IC by increasing the value units. If IC' isobtained from IC by increasing the value of more than
Lemma 1.2.7 Let in
vq
sequel n
one
=
a
(K, V2 )
=
curve
C
by increasing
weighted clusters going through 1C, going through IC2.
be two
of degree
virtual
values.
ones
d
and
Then: 1.
IC,
is obtained
where the ap
from /C2 by increasing the are given by
value
of each
p E K
by ap,
! 0
0r12
K
=
K
+
a
PEP
pEK
2.
We have the inclusion
of
linear systems
fr, (d)
C
r'2 (d) for
all positive
integer d. PROOF: The vector of effective values
VK(C)
definition of virtual transform
CfK(C) with
CiK(C)
=
(aP)pEK,
which
=
gives
P
1
2
0
assertion 1.
of C at K is V1.
By the
1 Preliminaxies
20
Since TF' > V2 ,
1.2.2
by
have for any
we
'HIC1,X
C
E
x
PI the inclusion of stalks
' IC2,X
and hence the desired inclusion of linear systems. 11 Next lemma shows how the excess, the virtual codimension and the orsingularity behave by increasing values, following directly from the
der of
definitions.
-
Lemma 1.2.8 Assume that the
weighted cluster IC'
from IC (K, v) of p, by n of excesses at p is by increasing pp' pp (rp + 1)n and at q 54 p, is either pq Pq -n if one of the points p or q is maximal among the points in K that are proximate to the other, or 0 otherwise. The variation of virtual codimension is is pq pq =
-
-
comes
=
units. Then the variation
the value
=
-
=
n
c(IC') and the variation
-
c(IC)
=
2
(2pp
+ 2 +
of order of singularity
(n
-
1)(rp
+
1))
1)(rp
+
1))
is
n
6 (IC,) _6 (IC)
.
2
(2pp
-
2 +
(n
+
of the cluster IC, and rp
where p,, is the excess at p, in K proximate to p.
is the number
of points
Enriques Q26] IV.II.17) called unloading (see weighted cluster IC (K, V) in S (K, v') is consistent and gives a new system of multiplicities v' so that IC' equivalent to IC. At each step of the procedure some amount of multiplicity /C < 0 from the points is unloaded on a point p, E K, at which the excess P that are proximate to it. Let us present it in terms of increasing values. There is
[5]
or
a
procedure
[6] 4.6)
that from
due to a
non-consistent
=
=
I
Definition 1.2.9 Let IC X
P
< 0 that
is, according
=
to
(K, v)
be
n as
the least
integer
weighted
cluster and
assume
that
1.1.44,
VtNKI-p Define
a
so
-n(rp
> 0
-
that +
1)
+
VtNK 1p
< 0
with rp the number of points in K proximate to p. the value of p by n.
Unloading
on
p, is
increasing
Assume IC (K, v) is a non-consistent weighted and, inductively, as far as IC'-' is not consistent define K' from )C'-' by unloading on a suitable point. Then we have: Theorem 1.2.10
cluster. Put ICO
=
([6] 4.6.2)
IC
=
1.2
There is
1.
K
as
that /Cm is consistent, has the
an m so
IC and is
Weighted
equivalent
same
clusters
21
cluster
underlying
to it.
only consistent weighted cluster which is equivalent to )C and underlying cluster. In particular, it does not depend on the of the points on which the unloadings are performed.
2. Km is the
has the choice
I et
same
introduce
us
Definition 1.2.11 tame
unloading.
K
as
rp!+_1
with 1 >
Example a
case
-1
=
of
unloading that
can
be found in
[6]
4.7.
point of excess equal to -11 will be called 1, unloading the value is increased in n and hence n is the least integer so that n > rp+l
Unloading
Note that in
-VtNKIp
=
P
special
a
a
on a
tame
(1-2.9). Figure 1.3 shows a sequence of three unloading steps from weighted cluster-to its equivalent and consistent one. Obsteps 1 to 2 and 2 to 3 are tame unloading, while the step 3
1.2.12
non-consistent
serve
that the
to 4 is not tame.
0 0
0
1
A-**
0 0
R3
R2
Fil
0
0
%0
0
0
F4]
Fig. 1.3. A sequence of unloading steps from a non-consistent system of multiplicities (left) to the,,, corresponding consistent one (right). The black indicate the points on which multiplicities are unloaded.
Proposition 1.2.13 ([6], 4.7.2) Assume from IC (K, v) by unloading on p E K. =
C(IC') and the
equality holds if and only if
Lemma 1.2.14 Assume that the
by
tame
unloading
in K. Then
on
:5 the
that the
weighted cluster IC'
virtual arrows
comes
Then
c(IC)
,
unloading
weighted
is tame.
cluster IC'
p, E K. Let rp be the number
comes from IC (K, v) of points proximate to p =
22
1 Preliminaries
J(r) and
if p,
is
a
-
non-maximal point
rp
(1. 1. 11),
J(r) ,
PROOF:
(1.5)
According to 1.2.11, pprby substituting in 1.2-8.
=
follows
then
Jpq
>
(1.5)
-
.
-1 and
n
=
1, after which equality
If p is non-maximal then rp > 1 and
hence the claim. 1:1
Remark 1.2.15 Let IC
(K, v) be a weighted cluster having non-negative multiplicities that gives rise to the consistent weighted cluster Q by tame unloading. The tame unloading steps may be performed in such a way that the intermediate weighted clusters have non-negative virtual multiplicities. Indeed, at each step, first drop successively maximal points with virtual multiplicity zero. Once there is no one of these, unload on a point that is maximal among those of virtual multiplicity zero if any. Since the amount unloaded at each step equals one (1.2.11), this guarantees that no multiplicity becomes negative. =
virtual
Lemma 1.2.16 Let IC
clusters in S
so
=
(K, v)
and V
c(/C') d(r) IV IV -
Furthermore, 1. IC
2.
=
-
be two consistent
weighted
the
following three
>
c(IC)
(1-6)
>
6(IC)
(1.7)
> Ic
Ic
-
assertions
(1-8)
are
equivalent:
V.
c,()C)
3. )c
(K, v')
=
that T' > -F. Then
IC
c(IC'). IV
-
V.
PROOF: Let p E S be
origin of
an
subclusters of IC and V whose
K(p) By 1.2..2, Wlcp,p
f1Cj}j=0'... np
D
=
K. Let
u pderlying
fq
and
)p
1CP'
be the
weighted
cluster is
E K: p, <
q}
C K.
Wic,"' p. Consider the flag of consistent weighted clusters withends Ko i.e. IC', )Cp and /Cn,, P
(see [5])
=
'H/c,,p and dim ?1jcj_1,p/W)cj,p
D
Wr,,,p
=
D
...
D
lir..',&
1 fo i 0,..., np* Ki is either the weighted clusQj obtained from 1Cj_j by adding a new infinitely near point of virtual multiplicity one if Qj results consistent, or is the consistent weighted cluster ter
=
=
1.3 Birational maps of surfaces
equivalent
Qi obtained by
to
if Qi results non-consistent.
unloading
tame
23
Therefore by 1.2.13 c
i
0,...,n
=
-
Oci+l)
()C')
=
+ 1
np >
+
c
p
we
(/Ci)
1, and thus c
If
C
=
unload
maximal
on a
point
c
(1-9)
(K)
origin of K
p of
Qi, then the multiplicity
at p
equals
1, which is negative. By 1. 2.15, the tame unloading steps leading the from Qi to ICi may be performed on non-maximal points of the intermediate excess
weighted
-
Hence, by 1.2.14,
clusters.
On the other hand 6
(Qi)
=
J
(/Ci+,). 6
(1.8)
Then
follows from
Xi)
(Qi)
6
(IC')
Thus
> J
(IC)
(1.3).
0 for each p origin of K, which only if np c (IC), and this proves the equivalent by (1. 9) to the equality c (IC') equivalence between assertions 1 and 2. Hence from (1.3) and (1.7) it follows the equivalence between assertions 1 and 3. 11 .Note that W
=
-9 if and
is
=
1.3 Birational maps of surfaces P
Let
:
S
--+
S' be
a
birational. map of surfaces.
Suppose S'
C I?n, To the
fixed part and map 4i we associate the linear system C = P*JHJ without dimension n, where IHI is the system of hyperplanes in pn The linear system .
C determines the map !P up to a projectivity of pn as there is a projectivity * + C *, 'with C* the projective so that u o!P is equal to the map S U : pn --+ C --
space dual to C, that sends divisors passing through x.
x
E S to the
hyperplane
in C
consisting of the
Definition 1.3.1 We will call C the linear system associated to weighted cluster of base points of C will be also denoted by IC(fl =
P. The
IC(C).
point x E S is a fundamental point of P if and only if point of the linear system C associated to P. Now let K S. Suppose SK C I n. By induction on the number OK of
Notice that the x
is
be
a
a
proper base
cluster in
blowing-upS", Let
us
the
recall
blowing-ups.
underlying cluster of
some
is K.
well-known facts about birational maps of surfaces and
24
1 Preliminaxies
Proposition 1.3.2 (Universal property of blowing up a point, [3] 11) Let f : X --+ S be a birational morphism of surfaces, and suppose that p E S is a fundamental point of the rational map'f -'. Then f factorizes as f where g is
birational
a
Lemma 1.3.3
-4 S, -4 S,
X
:
morphism and
([3] 11.9)
Let
f
:
X
e
--+
is the
S be
a
blowing-up of p.
birational
and suppose that p E S is a fundamental point Then the set f 1 (p) is a curve on X. Lemma 1.3.4
Q3] II.10)
Let !P
:
X
--+
S be
of
a
morphism of surfaces f
the birational map
birational map of surfaces
and suppose that p E S is a fundamental point of the birational map (P-1. Then there exists a curve C on X so that!P(C) = fpj. Theorem 1.3.5
([3] IIJI) Let f : S --* So be a birational morphism of surfaces. Then there exists a sequence of blowing-ups of points Ek : Sk --+ Sk-1 (k 0 1,...,n) and an isomorphism Uf : S --+ Sn so that f = Ej o =
En
...
0 U.
curve on
f : S --- S' be a birational morphism, f (C) jp}, with p point in S'. Then p is birational map f -1.
1.3.6 Let
Corollary S
so
point of the
that
=
and let C be a
a
fundamental
PROOF: Clear from 1.3-5. 11
f : S --+ S' be a birational morphism, and let K be a cluster in S so f =.UK o u, with u isomorphism (1. 3.5). Then the underlying cluster of
Let that
IC(f -1)
is K.
Theorem 1.3.7 Let 4i
IC
=
S
:
IQP)
that the
--+ =
(Universal property S' be
(K, v).
of
blowing
up
cluster)
a
birational map of surfaces, S' C pn, and put Then there is a birational morphism 77 : SK ---+ S' so a
diagram SK
"
(1.10)
45
S commutes and
f
:
Y
exists
below.
S/
the universal property that
for
any
pair
g
:
Y
--+
S,
S' of birational morphisms of surfaces fulfilling f 4i o g then there unique birational morphism h : Y '--+ SK commuting the diagram
--+ a
satisfies
>.
=
1.3 Birational maps of surfaces
25
Y h 9
f
SK K
S PROOF: Let
-
-
-
-
-
S/
-
first the existence of 77. Let C be the linear system
us see
on
S without fixed part associated to -P, i.e. C =!P*IHI, where IHI is the system of hyperplanes in pn D S'. According to 1.1.49 the linear system on SK
CK
(!P
=
HK)* IHI
o
VPEK
01C
=
:
C E C
P
pEK
has
no
fixed part and
morphism. The uniqueness of h it
on a
base
no
points. Hence
17
!P
=
17K is the desired
o
is clear because the commutative
diagram determines
dense subset of Y.
Note that if
fulfilling
=
g
we
17K
show the existence of
a
birational
morphism h
:
Y
-4
SK
h, then
o
f =(Pog=!PoJ-IKoh=77oh as
rational maps, and hence f = 77 o h as morphisms. For the existence of h induction on the cardinal OK of K.
we use
If
OK
=
1,
then K consists of
that p is a fundamental 1.3.2. We know that p is there is
a curve
Then the
curve
D
f
on -I
S'
(D)
commutative relation
one
point of the a
proper base
map
point
g-1, then
p
only. If
we
show
the claim follows from
fundamental point of the map (P. Hence from 1.3.4 that its image by the map 4i-I is 4i-I (D) = jp}.
so
on
P-'
Y maps
=
g
o
by g to the point p E S, owing to the f -1. Thus, according to 1.3.6, g-1 has p as
fundamental point. If OK > 1, write K
the points numbered following an admissible total ordering. The base point pi must be a proper point in S, and hence pi is a fundamental point of the map (P. Reasoning as in the case OK 1, it follows that p, is a fundamental point =
of the map h, : Y -+ the
point
g-1. Applying 1.3.2, S1,
pl,
with
so
there exists a birational morphism of surfaces S, the surface obtained from S by the blowing-up ep, of
that the
diagram hi
S,
Y
If
9
EPI
S
-
-
-
-
-
B.-SI
1 Preliminaries
26
commutes. Then consider the linear
C, with
Ep,
P,*JHJ
:=
system
=
e;Pi C
S,
on
-
jLp1EPI
exceptional divisor of the blowing-up Ep,, and Pj weighted cluster IC, (KI, p) in S1, with
the
Observe that the
o
epl.
=
K1
=
JP2)
)PC})
...
is the cluster of base
induction
f : Y -+ S' diagram
and
h,
points of the linear system C1. Thus we can apply 4i, and the birational morphisms Y Sj that satisfy the commutative relations of the
to the birational map
hypothesis :
Y '
\\ 915 1
S, Then there exists
a
birational
-
-
-
-
-
,
-
S1
of surfaces h
morphism
Y
SK
so
that the
diagram h
SK
Y hi
HK1
f 951
S/
commutes, where ITK, is the blowing-up of the cluster KI. Wehave
ITK Thus h is the birational
'Corollary IC
=
IC(fl
morphism
1.3.8 Let 4i
(K, 1L)
=
=
:
S
and L
--+ =
HK,
0
Ep,
we were
S' be
IQV1)
looking
a =
-
for. El
birational map of surfaces, put The birational morphism
(L, v).
: SK -+ S' given in 1.3.7 is the composite of a uniquely determined isomorphism u : SK. ---+ SL and the blowing-up HL : SL -4 S' of the cluster
,q
e.
n
PROOF:
morphism
Applying SL
---+
11L
0 U.
1.3.7 to the birational map so that the diagram
!V',
there is
a
birational
S
SL
S commutes and
(SL, , HL) satisfy the
Therefore there exists
a
S1
same universal property as (SK, -UK, 77). unique birational isomorphism. u so that the diagram
1.3 Birational maps of surfaces
27
SK
I -UK
SL
S
-
-
-
-
-
S/
-
commutes. r-1
Definition 1.3.9
Keep the
section in S' and C
notations of 1.3.8 and let H be
a
hyperplane
4i* (H). We say that C and H correspond to each other A. Observe that if we identify the divisors on SL with their isomorphic
by images
=
SK, then
on
Corollary
1.3.10 Let
P
:
S
S' be
--+
birational map of surfaces, and of 1QP) and L is the underlying morphisms of surfaces so that the
a
'
suppose that K is the underlying cluster cluster of IC(fl. Let g and f be birational
diagram Y
I
/ A
S
Suppose that K' is underlying cluster of IC(f -1). commutes.
-
the
-
L C L'
-
we
---+
S/ is
the
have the inclusions
of clusters
in in
S,
S',
equality of subsets
are
=
L'-L,
clusters in SK.
PROOF: In virtue of 1.3.7 there exists Y
>-
of clusters
K'-K which
-
underlying cluster of IC(g-') and L'
Then
K C K'
and the
-
SK
so
that the
a
unique birational morphism h
diagram Y h 9
SK
f
-UK
'I--
S
S/
1 Preliminaries
28
Q be the underlying cluster of IC(h-1) 1.3.8,
commutes. Let
According
to
on
the surface SK.
K'=KUQ, L'= L U Q, and the-claim follows. Lemma 1.3.11 Let
P
IC(4i)
=
=
and C
(K, 1L)
S, C pn, and take p,
S
:
S' be
--+
IC(4i-')
E K.
birational map of surfaces, K Let H be a hyperplane section in
a
(L, v).
=
Then E K.
HL
=
. 0.
JLP
P
PROOF: Let C be the linear system associated
01"
HL (1.3.9).
=
Then
to!P, and let C
E C
so
that
by 1.1.26, -L
-==K
EP
,
H
-K =
-
Ep-
C
-
IC
Itp > 0.
=
EI
Proposition 1.3.12 Keep the notations and hypotheses of a hyperplane section in S' C pn, and take p, E K'. Then ==K'
EP_
1.3.10. Let H be
-_L'
H
-
> 0.
-K'
Furthermore, PROOF:
0
Ep-
if and only if p,
=--L'
K,
then E
L' '
H
=
h*
P
If p, E K- K
=
(--K E_ P)_ h*
1.3.13
-:-L'
EP_
HI
3.E q
following
surfaces
-K'
L'
Keep the
K =
ITK'
Y and
> 0 > 0
for
for
PROOF: It follows
SK
H
=
ILV > 0
substituting
in
(1.11) gives
--L' -
H
conditions
all the
L ,
P
0 and
o.
notations and
are
E
K
(
are
isomorphic.
points
p E K.
all the points q c L'.
directly from
Ei
hypotheses of 1.3.10. a hyperplane section equivalent:
section in S C F' and let H' be
Then the three
2.
H > 0
(_HL)
f,, FP
L'- L, then
Ep-
The
K.
K
(EP-
H
-K'
1.
-
1.1.26 and 1.3.11
by
K'
Corollary hyperplane
KI
By the projection formula (1.1.26, 6)
EpIf p E
E
1.3.12. El
Let H be
a
in S' C Pn.
