Resolution of Singularities

V,le can extend our definition of a resolution of singularities to arbitrary schemes. A reasonable category to consider is excellent (or quasi-excellent) 8chelnes (defined in IV.7.S [45J and on page 260 of [66]). The definition of excellence is extremely technical? but the idea is to give minimal conditions ensuring that the the singular locus is preserved by natural base extensioTIfoi such as cOInpletion. There are examples of non-excellent schemes 'Which admit a resolution of singularities. Rotthaus l74] gives an example of a

}O

2. Non-singularity and Resolution of Singularities

regular local ring R of dimension 3 containing a field which is not excellent. In this ease; spec( R) is a resolution of singularities of spec(R) .

2.4. Normalization Suppose that R is an affine domain with quotient. field L. Let 8 he the integral closure of R in L. Since R is affine, S is finite over R, so that .~y is a,fnne (cr. Theorelll 9, Ohapt.er V [92}). We say that spec(S) is the normalization of spec( R) . Suppose that V is a valuation ring of L such that ReV. V is integrally closed in L (altllOugh V need not. he Noetherian). Thus S c V. The morphism spec(V) -+ spec(R) lifts to a morphislll spec(V) ---+ spee(8), so t.hat. spec(S) .) spcc(R) is proper.

If X is an integral variety, we can cover .x by open affines spec(Ri) with normalization spec(Si). The spec(Si) patch to a variety Y called the normalization or x. Y ----t X iR proper, by our local proof. For a general (reduced but not necessarily integral) variety X, the normalization of X is the disjoint union of the normalizations of the irreducible components of X. A ring R is normal if Rp is an integrally closed domain for all p E spec(R). If H. is an affine ring and spec(S) is the normalization of spec(R) we construeted above, then 8 is a normal ring.

Theorem 2.13 (Serre). A Noetherian ring A is normal if and only if A is Rl (A p ,is 'T'eg1J.lm' -if ht(p) < 1) a.nd 82 (depthAp > min(ht(p),2) for all P E spec(A)). A proof of thi~ theorem is given in Theorem 23.8 [66].

Theorem 2.14. Suppose that X is a l-dimensionc~l vadety o'Ue',. a, jteld K. 1'Ium, the normalization of X is a resolution of singularities.

Proof. Let X be the normalization of X. All local rings 0x,p or points p E X are local ring" of dirnelli3ion 1 which are integrally closed, ~o Theorem 2.13 inlplies they arc regular. 0 As an example, consider the curve singularity y2 - :1;3 = 0 in A2 , with affine ring R = K[x, y]/(y2 - x:l). In the quotient field L of R we have the relatiOJI (~)2 - :1; - O. Thus ~ is integral over R. Since R[~] = K[~] is a regular ring, it must be the integral closure of R ill L. Thus spee(R[~]) ---+ spec(R) is a resolution of singularities. Normalization is in genera] not enough to resolve singll.laritie..<; in dimension larger than 1. As an example, consider the surface singularity X defined

2.5. LocaJ uniformiza,tioIJ and generalized resolution prohlems

11

by z2 - xy = 0 in A=J<, where K is a field. The singular locus of X is defined by the ideal J = (z2 - xy, y, x, 2z). VJ = (x, y, z), so the singular locus of X is the origin in A2 . Thus R = K[x,y,z]/(z:.! - xy) is R I . Since z2 - xy is a regular element in K[x, y, z], R is Cohen-r\'lacaulay, and hence R is 82 (Theorem 17.3 [66]). We conclude that X is normal, but singular.

Kawasaki [58] has proven that under extremely mild assumptions a Noetherian scheme admits a Macaulayfication (all local rings have maxiIUal depth). In general, Cohen-M.aeaulay schemes are far from being uonsin~ular, but they do share many ~ood homolo~ical properties with nOnsingular schelnes. A scheme which does not admit a resolution of singularities is given by the example of Nagata (Example 3 in the appendix to [68]) of a onedimensional Noetherian domain R whose integral closure R is not a finite Rmodule. We will show that such a ring cannot have a resolution of singularities. Suppose that there exists a resolution of singularities ¢ : Y - t spec(R). That is, Y is a regular scheme and 4) is proper and biralional. Let L he the quotient field of R. Y is a normal scheme by Theorem 2.13. Thus Y has an open cover by affine open sets [h, ... , Us such that [h ':Vspec(li), with Ti = R[y'i1~ ... , YU] where YijE L~ and eaeh T, is integrally dUl:led. The inte~ally closed subring s

T=r(~Oy)=n1i i=I

of L is a finite R-module since ¢ is proper (Theorem III, 3.2.1 [44] or Corollary II.5.20 [47] if cP is assumed to be projective).

Exercise 2.15. 1. Let K = Zp(/;), where p iR a prime and l is an indeterminate, I.. et R = K[x, y]/(xP + yP - tL X = spec(R). Prove that X is nonsingular, but there are no points of ,x. which are smooth over K.

2. Let K = Zp(t), where p > 2 is a prime and t is an indeterminate. Let R = K[x, y]j(x 2 + yP-- t), X = spcc(R). Prove that X is nun-singular, and X is smuoth overK at every point except at the prime (yP - t, x).

2.5. Local uniformization and generalized resolution problems Suppose that f is irreducible in K[[x, y]]. Then the Puiseux series (2.5) of Sect.ion 2,1 determines an inclusion

(2.7)

H = K[[x, y]]j(f(x, y»)

-t

K[[t]]

12

2. Nun-."ingularity and Resolution of Singularities

-----------------=--

with :r. = tm., y = p(t) the !Series of (2.5). Since the 'Tn chosen in (2.5) of Section 2.1 is t.he smallest possible: we can verify that t is in t.he quotient field L of R. Since the quotient field of [<[[iJl is generated by 1, t, ... ,t.m 1 over [~, we conclude that Rand K [[t]] have the same quotient field. (2.7) is an explicit I't~alization of the nonnalh"a,tion of the curve singularity germ f(x, y) = 0, and thulS :;pe<.~(K [[t]]) ~ spec(R) ilS a resolution of singularities. The quotient field L of ]([[t]] has a, valuation v defined for non-zero h E L by v(h(t)) = n E Z

il' 11,(/.) = t71. u , where u is a UIIit, ill K[[t]]. This valuatioll ean be understood in R by the PuiselDC series (2.6). ~Iore

generallYl we can consider an algebraic funct.ion field L over a field 1<. Suppose that V is a valuation ring of L containing K. TIle problem of local uniformizatiofl is to find a regular local ring R, essentially of finite type over K and with quotient field L such that the valuation ring V dominates R (R c V and t.he intersection of the maximal ideal of V with R is the maximal ideal of R). The exa.ct relationship of Ioeal uniformization to resolution of :;ingulfllitics is explained in Section H.3. If L has dimension lover K, then the valuation rings V of L which contain 1< are predlSely t.he local rings of points on the unique non-singular projective curve C with function field L (cf. Section I.6 [47]). The Newton method or Section 2.1 can be viewed as a, solution to the local unirormization problem for complex a.nalytic curves.

IT L ha..., dimcnsion > 2 over K, t.here arc many valuationR rings V of L which are non-Noetherian (see the exercise of Section 8.1). Zariski proved local ulli 1'0 rrnization for two-dirneHsional function field:; over an algebraically closed field uf characteristic zero in [86]- He was able to patch together local solutions to prove the exi'1tence of a resolution of si nglliarities for algebraic surfa.ces over an algebraically dosed ficld of characteristic ~ero. He later was able to prove local uniformization for algebraic fmlction fields of cltaracteristic zcro in [87]. This method leads to all ext.remely difficult patching problenl in higher dimensions, which Zariski was able to solve in dimension 3 in 190]. Zarisk i:s proof of local uniformization can be considered eu; an extension of the Newton method to general valuations. In Chapter 8, we present. Zariski's proof of resolut.ion of surface singularities through local uniformizatioll. Local uniformizatioIl ha.~ been proven for two-dimensional function fields in positive eharacteristic by Abhyankar. Abhyankar has proven resolution of singularities ill pOl;it.ive characteristic for surfaees and for three-dimen:;ional varieties, [1] ~ [3J, [4].

2.5. Local Ilnifonnizatio11 and generalized resolutio11 problems

There has recently been a resurgence of interest in local mliformization in positive eharacteristie, and significant progress is being made. Some of the recent papers on this area are: Hauser l50j, Heinzer, Itotthaus and Wiep;and [51J, Kuhlmann [59], [60], Piltant [71], Spivakovsky [77], Teissier [78]. Some related resolution type problems are resolution of vector fields, resolution of differential forms and monomialization of morphisms. Suppose Lha.L. X is a va.riety whir:h is smooth over a p~r1eet fi~ld K, a 1111 D E Hom(Ok/K' Ox) is a vector field. If p E X is a closed point l and Xl, ... : X n are regular parameters at p, then there is a local expression

D=

u () a1 ( x) 8 + ... + an (1:) () , Xl Xn

where ai(x) E OX,p. We can associate an ideal sheaf ID to D on X by

for [J E -"Y. There are an effective divisor Fn and an ideal sheaf ~T1J such that V(\.1D) has eodimem~ion > 2 in X and In = CJx( -F'n)Jn. The goal of resolution of vector fields is to find a proper morphism rr : Y --+ X so that if D' = rr- l (D), then the ideal ID' C Oy is as simple as possible. • ThiR waR accomplished by Seidenberg for vector fields on non-singular surfaces over an algebraically closed field of characteristic zero in [75]. The best statement that can be attained is that the order of J D' at p (Definition A.17) is Vp(JDI) -::: 1 101' all p E Y. The basic. invariant. considered in the proof is the order lJp(Jv). vVhile this is an upper semi-continuous function on Y, it can go up under a monoidal transform. However, we have that VIl(JDI) 5 Vp (JD)+l il'7r: Y --+ X is the blow-llpof]J~ and q E 7r- 1 (p). This should be cOlnpared with the classical resolution problem, where a basic result is that order cannot go up under a permissible mOlloidal transform (LeIlllua 6.4). Resolution of vectur fields for SlIlooth SUl'ra<:es over (1, perreet field has been proven by Cano l19J. Cano has also proven a local theorem for resolution for vector fields over fields of characteristic zero [20J, wllich implies local resolution (along a valuation) of a veduI' field. The statement is that we can achieve lJq(JD/) < 1 (at least locally along a valuation) after a morphism 7r : Y --+ X. rrhere is an ana.logolls problem for differential forms. A recent paper on this is [21.]. We can also consider resolution problems for morphisms f : Y --+ X of V'dIieties. The natlU'al question to ask 1s if it is possible to perforrn rnonoidal transfonns (blow-ups of non-singular subvarieties) over X and Y to produce a morphism which is a monomial mapping.

14

2. Non-singularif,y and Resolution of Singularitie>

Definition 2.16. Suppose that : X

~

Y is a dominant morphisrn of lLOIIsingular irreducible K-varieties (where K is a field of characteristic zero). q> is monomial if for all p E X there exists an etale neighborhood U of p, uniformizing parameters (Xl,'" ,X n ) on U, regular parameters (YI,· .. ,Ym) in OY,I!>(p) , and a matrix (aij) of non-negative integers (which necessarily has rank m) such that

Ym

·• ·

,.,.aml . . . xO"mn

=

·j'l

n'

Definition 2.17. Suppol:le that : X ~ Y is a dominant morphism of integral K-varieties. A morphism 'IT : X 1 ~ Yj is a monomialization of Q; :

[27} if t.here are sequences of blow-ups of non-singular subvarieties Xl ~ X and /3 : Yi ~ Y, and a morphism '11 : Xl ---I> YI such that the

cl)

diagrarn

Xl

1 X commutes, and W il:l a rnollc)]uial morphism.

X ~ Y is a dominant morphism from a 3-dimensional variety to a surface (over an algebraically closed field of chanu.:teristic 0), then there is a monomialization of [27]. A generalized multiplicity is defined in this paper, and it can go up, causing a very high complexity in the proof. An extension of this result to strongly prepared morphisIIlS from n-folds to surfaces is proven in [33]. It i~ not. known if nlonornialization is true even for birational morphisms of varieties of dimension > ~{, although it is true locally along a valuation~ from the following Theorem 2.18. Theorem 2.18 is proven when the quotient field of S is finite over the quotient field of R in [25}. The proof for general field extension~ is in [28].

Tf

:

Theorem 2.18 (Theorem 1.1 [26], Theorem 1.1 [28]). Suppose that R c 5' UTe. Tegu.lar Local rings, essentially of finite tYlle mler a field K of characteristic zero. Let V be a valuation ring of K 'Which dominates S. Then there exist sequence.s of monoidal transforms R -+ R' and 8 ---I> S' .'1'Uch that V dominates 8', 8' dominates R' and there are regular parameters (XI, .... , a: m ) in R', (Yl, ... , Yn) in 8', units 81 , ... ,8m E 8' and an m x n matrix (aij) of non-negative. integer8 .'1'Uch that rank( aij) = m is maximal and Xl

·••

(2.8) Xm

2.5. Local uniformizatioJ1 and genel'ali7Jed resolution problems

15

Thus (since char(K) = 0) there exists an etale extension 8' ~ S''', where 8" ha~ regular pararnetersfj 1, ••• ,fin such that XI,· .. ,Xm are pure monomials in Yll ... ,Yn · The standard theorems on rp,solution or singularities allow one tu e~ily find RJ and SJ such that (2.8) holds, but, in g;eneral, we will not have the essential condition rank( aU) = m. The difficulty of the problem is to achieve this condition. This result l!;ives very simple structure theorems for the ramification of valuations in characteristic zero function fields [35]. We discuss some or these results in Chapter 9. A generalization of monomialization in characteristic p function fields of algebraic surfaces is obtained in [34] and especially in [35]. ~Te

point out that while it seems possible that Theorem 2.18 does hold in positive characteristic, there are simple examples in positive characteristic where a. monomia.li~a.L.ion does nol exi~t. The siIllplest example is the nlap of curves

in characterist.ic p. A quasi-complete variety over a field K is an integral finite type Kscheme which satisfies the existence part of the valuative criterion for properness (Hirona.ka., Chapter 0, Sectiun 6 uf [521 and Chapter 8 uf [26]). The construction of a monomialization by quasi-cOlDplete varieties follows from Theorem 2.18. rl'heorem 2.19 is proven for gflnerically tini1.fl mol'phisTIls ill [261 and for arbitrary morphisms in Theorem 1.2 [28J.

Theorem 2.19 (Theorem 1.2 [26],Theorcm 1.2 [28]). Let K be a. field of (~ha,r(ldt~1'ist'iG ZtTO, : X ~ Y (J, dumiu,ant mU'fphism of prvper' K -varieties. Then there exist birational morphisms of non-singular quasi-complete Kvarieties Q: : X1} X and (J : 1'1 Y! and a monom:ial l1uwphi.'1111. l1J : Xl ---j. Y1 ,';'lu:h Uwt the diagram -4

commutes and a and (3 are locally products of blow-ups of non'-singular subvarieties. That is, for every Z E Xl, there exiofit afJi1u~ neighborhood.., V, of z and V ul:l~ = o-:(z) such that 0: : Vi V is a finite product ofmonoidal transforms~ and there exist affine neighborhoods WI of 'J!(z) , W of y = p('J!(z)) such that fJ : W1 ~ W is a finite produd of mono,i.dol lra.7MJjo17Tl.S. ---j.

16

2. Non-sillgularity and R.esolution of Singularities

A mnnnidaJ trfLllsfornl of a non-singular K..;scheme S is the Dlap T - t S induced by an open subset T of proj(ffiP), where I is the ideal 8h~ar of a non-singular subvariety of S.

The prool' of Theorem 2.19 foliowR frOln Theorem 2.18, by patching a finite number of local solutions. The resulting schernes luay not. be separa.ted.

1t is au extremely interesting question to determine if a monomialization exists for all morphisms of varieties (over a. Held or eharader·iRt.ir: zero). That. is, the conclusions of Theorem 2.19 hold, but with the stronger conditions that. (): and [J are products of monoidal transforms on proper varieties Xl

and Y I .

•

••

......... Chapter 3

Curve Singularities

3.1. Blowing up a point on A2 Suppose that]( is an alg-ebraically closed field. Let

Define a K-a.lgebra isornorphisrn

,X : K[s, t]", = K[s, t, 1] M

by A(.S) - ~: ,XU)

=

'WI!.

.~.~ [([u, tllv = K[u, tI, 1] V

We define a K-variety Eo by patching [h to lh

OIl

the open sets

U2 - V(v) = spec(lflu, vJv) and U1 - V (8) = spec(K[s: tl s ) by the isomorphism A. The K -alp;ebra hOlnomorphisms

[
f---*

st, y

f---*

t, and [([x, y]

defined by x 1-·) 11., Y get a 1I1OrphisIIl 7r :

1-)

B(p)

- [([s, tJ

UV,

-+

K[u, v]

arc compatible with the isomorphism ,x, so we

= Bo ~ Uo = spec(K[x, y]) -

Ak:

where p denotes the origin of Uu. Upon localization of the above maps, we get isomorphisms

K[x, Y]lI

c::

K[s, tlt and K[x, Y]x

~

K[u, v]u. 17

8. Curve Singularities

18

Thus 1r : VI - Vet) ~ Uu - V(y), 7r : U2 - V(u) c: Uo - V(x), and isomorphism over Un - Vex, y) = Uu - {pl. Moreover, 1r1T-

1

1

7r

is an

(p) nUl = V(st, t) = {V(t) c spec(K[s, t])} = spec(K[s]),

(p)

n U2

c

= V('U,U'lJ) = {V(u)

1 spec(K[u,vD} = spec(Kr-l),

s by the identification /:I =- ~. Thm; 7[-1 (p) '" pl. Set E = 1T- 1 (p). We have "blown up IJ" into a codimension 1 subvariety of B(p), isomorphic to ]pl. Set R = K[x, y],

Tn

= (:v, y).

Ul = spec(I([s, tD

U2

Then

= spec(K[xy y]) = spec(R[x]), y I

= spec(K[;J;,

y _cD = spec(R[-'Y D. a: x

We see that

B(p)

x Y = spec(R[-]) U spec(Rr-l) = proj(EB 1nn ).

x

y

.

1/.>0

SuppORe that q E 7f- 1 (p) is a closed point. If q E Ub then its associated ideal m q is a maxirnal ideal or Rl = K[~ I y] which contains (x, y)Rl - yRl' Thus m q = (Y, ;, - 0) for smne (}J E K. If we set Yl - y, Xl = ~ - (\I, we see

that there are regular paramel.erfi (XI, y,) in 0 B(p),q such that

x = Yl (Xl

+ Q),

Y

= Yl·

By a similar calculation, if q E 7f 1 (p) and q ( U2 , there are regular paramet.ers (a;" Y I) in 0 B(p),q such that

x =-

+ ;3)

Y = Xl (Ul

:1:1,

for sorne lJ E K. If the constant Q or (3 isnoIl-zero, then q is in both Ul and U2. Thu8 the poillts in7r- 1(p) can be expressed (uniquely) ill oue or the forms

x

=

Xl,Y ~ a:l(Yl

+ (~)

with

or x = XIYl,Y = Yl-

0: E ](,

Since R(p) is projective over spec(R) , it cerbl,inly is proper over spec(R). However, it is illurn i na.ting to give a direct proof. Lemma 3.1. B(p) ~ spec(R) is proper.

Proof. Suppose that V is a valuation riug containing R. Then ~ or ~ E V. Say ~ E V. Then R(~] c V, and we have a morphism

spec(V)

spec(R[l!.]) C B(p) x which lifts the morphism spec(V) --+ spec(R). --+

I

0

3.1. Blowing up a point on A2

19

More generally, 8Upp0l=le that S = spec(R) is an affine surface over a field L, and pES is a non-singular closed point. After possibly replacing S with an open subset spec(Rf), we may assume that the maximal idea.! of p ill R is rnp = (x, y). We can then define the blow up of pinS by

(3.1)

7r :

B(p) = proj(EB TTl';') ~ S. on>O

We can writ.e B(p) as the union of two affine open subsets:

D(p) = spec(R[x)) U spec(R[YJ). y

7r

is an isomorphism over S - p, and

:r.

1r-

1 (p)

f"V

pl.

Suppose that S is a surface, and pES is a non-singular point, with ideal sheaf m,p cOs. rrhe blow-up p E 5' is

or

?f ;

B (p)

= proj (EB 111~) ---+ 8. n>O

is an isomorphisnl away from p, and ir u neighborhood of p in S such that

1f

(x, y)

=

~

spee(R) c S is an affine open

r(U, 17~p) c R

is the maximal ideal of p in R, then the map the construction (3.1).

7r : 1r -1 ( U) ---+

U is deB ned by

A1<

Suppose that C c is acurve. Since K[x, yJ is a unique factorization domain, there exLc;ts l E: K [x, yJ snch that. V(J) = C.

If q E C il=l a closed point, we will denote the correspondinl?; maximal ideal of K[x, yJ by m q • Wc*set

vq(C) = maxfrl f

E "'L~}

(more generally, see Definition A.17). vq(C) is both the multiplicity and the order of C at q. Lemma 3.2. q E

Ak i.l:J a nmz-.9ing'lLlm· l)(rint of C

'if unci only if IIq (C)

= 1.

Proof. Let R = ]([x,Y]mq, and suppose that mq = (x,y). Let T = RIIR. 1£ vq(C) > 2, t.hen f E m.~, and TnqTlm~T ~ mq/,m~ has dimeul:liQll

2 > 1, so that T is not regular and q is singular on C. However, if 'lIq (C) = 1, then we have f (~X + fry mod 'ln~,

=

where Q, (3 E ]( and at least one of a and 13 is non-zero. Without loss of generality, we may suppose that fJ ¥= O. Thus

Y-

-~x

mod

m~ + (I),

20

3. Clll've Singularities

and x is a !( generator of 'TnqT /,m~T

m,q/m,~ + (/). We sec that

f".J

dimK 'lnqT/1lt~T = 1

D

q is non-Ringular on C.

RTld

Remark 3.3. This lemma is also true in the situation where CJ is a nonsingular poi nt on a non-singular surface S (over a field !() and C is a curve contained in S. We need only Illodify the proof by replacing R with the regular local ring R = OS,q, which has regular parameters (j\ y) which are a K(q) basis 0/' m,q/m,~, and since R is a unique factorization domain, there is .f E R such that 1=0 is a local equation or C a.t q.

Ak,

Let p be the origin in and RUppOSC that vp(C) = r > O. Let 7T" : B(p) ---+ A'k be the blow-up of p. The strict transform 6 of C in B(p) is the Zariski closure of iT- 1(C - p) ~ C - p in B(p). Let E = 7r-1(lJ) be the exceptional divisor. Set-theoretically, 7[-1 (G) = 6 U E. For some

aijE

K, we have a finite sum

L

f =

aijXiyi

i+j>r

=I

0 for SOrIle i, j with i spec(K[xl, Yl]) of B(p), where

+j

with

aij

Xl =

0 is a local equatioJl for E. Also,

f where

/J

=

=

L

"=

In the open subset

'f'.

U2

xl!l, • r (Lij:ci+J -

Yi .

i I j>r

f !l

since vp (C) = r, so that 11 transform C of C in U2 . Xl

Tn t.he open subset U1

0 is a loeal equatioll

= spec(K[xl, jhD of D(p), where x

IiI =

=

=

Xl Yl ,

Y = Y 11

0 is a local equation for E and

f where

is a local equation of C in Ul.

=

yj]l'

or

th e strict.

3.1. Blowing up

Ji

point

21

OJ} A,2

As a scheme~ we see that 7T- 1 ( 0) = C) + 1'E. The Rcheme-theoretie preimage of C is called the total transform of C. Define the leading form of .f to be j L = aijx'i y •

L

i+j=r

Suppose that q E 1i- 1(p). There are regular parameters

(Xl,

yt) at q of

one of the fonnA :1: = :tJl'!}

In the first case f1 =

= XdYl + 0:)

or ~r. =

XIYl,

Y=

Yl·

!r = 0 is a local equation of 6 at q, where il = L aij(YI + o)j + x1n 1

i+i=r for' some polynomial H. Thus vq(C)

2:

< 'f', and vq(C)

ai.i (Yl+~ a:)j

=

'f'

implies

= aOr Vl.

i+.i=r

'\iV'e then see that

L

aijy{ = aUr(Yl - o:)r

i+j-r

and L = ao", (y - ax)'/'.

In the second ca~e

II =

-4 = 0 is a local equation of {; at q, where Yl

L

11 =

llij2:;

+ YIn

'i+j=1'

for some series

n.

-

Thus lIq(C) <

T,

~

-

and vq(C) = r implies 'I,

L"

aijx\

=

'/'

arUxl

i I j=r

and

L

= arox r .

We then see that there exists a point q E 1r- 1 (p) such that vq(C) = r only when L = (ax +by for some constants u, bE K. Since r > 0, there is at most one point q E 7r 1 (p) where the multiplicity does not drop.

r

Exercise 3.4. Suppose that Klxl, ... , xn] is a polynomial ring over a perfect field K and f E K[Xl,'" )xn.l is such that. !(O, ... , 0) = O. Let. rn = (Xl, ... , X n ), a maxiInal ideal of K[ Xl, ... ,xn ]. Define ord !(O, ... ,0, x n ) = r if x~

I f(O, ... , 0, x n )

and x~+l t /(0, ... ,0, x n ).

22

3. Curve SingulfLrities

Show that. R = (I([xl, ... ,xn,]/(f))m orr! 1(0, ... , 0, x n ) = 1.

IS

a. regular local ring if

3.2. Completion Suppose that A is a local ring with maxirnal ideal m. A coefficient field of A is a subfield L of A which is mapped onto Aim by t.he quotient mapping A --) A/m. A basic theorem of Cohen is that an equicharactcristic complete local ring contains a. coefficient field (Theorem 27 Section 12, Chapter VIII [92]). This leads to Cohen's structure theoreUl (Corollary, loco cit.), which shows that all equicharacteristic eompletc regular local ring A is isomorphic to a formal power series ring over a field. in fact, if L is a coefficient field of A, and if (Xl"", Xn) is a regular systenl of parameters of A, then A is the . . power senes rmg .

A

=

L[[Xt, ... , xn]]·

We further remark that the completion of a local ring R is a regular local ring if and only if R is regular (cf. Section 11, Chapter VIII [92]). Lemma 3.5. Suppose that 5' is a non-sin,gular algebraic surface defined over a field ](, p E ., is a closed point, 7r : B = D(p) -+ S is the blow-up of p, and ,s7/,ppose that q E 7r l(p) is a closed point .~1J,ch that I«(q) is .':ttlJarable over K(P). Let R 1 = OS,p and R 2 = OB,q, and suppose that K J is a coefficient A

"

field of Rl (x, y) aT'e 'regular parameters in RJ • Then there exist CL r:oefficient field K 2 = K 1(0:) of R'2 and regular parameters (Xl,vJ) of R2 such thai. l

n-* : H) i.~

--t

R2

the map given by

2: 'i,j>O

where

aij

E o.ijX;+i (Yl + o)i,

aijXiyJ -+

iJ>O

E Kl' or K2 = K 1 and

it'" : R 1

-+

R2

is the map given by

Proof. We have R1

~

K1 [[x, yJ], and a natural homomorphism Os

-+

OS,p

-+

RI

23

3.2. Completion

which induces a q E B(p) x.c; SpCC(Rl) = spec(Rl [Y]) U spec(Rd :]).

y

~c

Let mq be the ideal of q in B(p)x.c; spec(Rl). If q E spec(Rd~]), then

K(q)

::i

Rl[~l/1nq ~ Kd~JI(f(~))

for some irreducible polynomial l(¥) in the polynomial ring Kl[~]. Since K(q) is separable over K 1 ~ K(p), f is separable. We havem q = (x, l(~)) C

R 1 [~]. Let a E Rl [~] /1n q be the cla..'is of ~. We have Lhe r'esidue Inap 1> : R'2 -+ L where IJ -= Rd~1I1nq. There is a natural embedding of Kl in R2. Vv'e have a factorh'.a,t.ioll uf f(t) = (t - a)')'(/,) in D[',], where t - it and ,(t) are relatively prime. By Hensel's Lemma (Theorem 17 7 Section 7, Chapter VIII [92]) there is a E R2 such tha.t 4>(0') = a and f(t) = (t - ah,(t) in R2[tL where (/J(,(t)) - ')i(t). The suhfieJd K2 - K 1 [0'] of Hz is thus a coefficient field of R21 and mqR2 = (x, ~ - a). Thus R2 = K2[lx, ~ - a]]. Set Xl = X, Yl a. The inclusion

=;

Rl = ](1 [[x, y]]--+ R2 = K2 [[Xl! y,]] is natural. A series

L:aiixiyi with coefficients aij E Kl maps to the series

L: aijx'i+:i (Yl + a)j. o We now give an example to show that even if a regular local ring contains a field~ there may not be a coefficient field of the completion of the ring containing that field. Thus the above lemma doe~ not, ext.eTld to non-perfect fields. Let K = Zp(t), where t is an indeterminate. Let R = K[xkxp-t). Sup. '" " pose that R has a coefficient field L containing K. Let ¢ : R --) R/1n be the residue map. Since ¢; I L is an isomorphisln, there exists A E L such that '" >..P = t. Thus (x P - t) = (x - >..)P in R. But this is impossible, since (;r,P - I.) is a. generator of mR, and is thus irreducible. ~

Theorem 3.6. Suppose that R i.'l a fp.d'lLCp.d affine ring ove7' lL field K, and A = Rp, wher'e IJ is a prime ideal of R. Then the completion A. = Rp of A with respect to its m.aximal ideal is reduced.

3. Curve Singularities

24

When I{ is a pel'feet field, this is a theorem of Chevalley (Theorem 31, Seetion 13. Chapter VIII [921). The general case follows froIll SdlUlie IV 7.8.3 (vii) [45]. However, the property or being a donlain is not pre:served under complet.ion. A simple example is f = y2 - :1: 2 + ::r3 • f is irreducible in C[:c, y], but is reducible in the completion Cllx, yJ]:

/ = y2

x 2 - x 3 = (y - xV1

+ ;r,)(y + xVI + x).

or the two factors y-x- 1 x 2 + ...

The first parts of the expan:siuws

are

2

and

ytx+ 1x 2 +....

2 Lemma 3.7 (\Veiel'stra."s Preparation Theorem). Lei, K be a field, and ffIJ-l)pnSC

that

f

E

[<[{Xl, ... ; ;Tn,'!}]] i.9 such that

o< r = 1/(/(0, ... ,f), y)) =

max{n I yn divides f(O, ... , O~

Then I.he1'f~ e.rist a unit series u in I<[[:q, ... , Xn., ai E

yn < ce.

y]] and non-unit ser'ie.s

]
.f =- u(y'" + aly'r-l + ... + al')' A proof is given in Theurem 5, Section 1, Chapter VII [92]. A cuncept which will be important in this hook is the Tschirnhausen transformation, which generalizes the ancient notion uf "eompletion of the sqnare" in the solution of quadrati<: p.qllatiollS.

Definition 3.8. Suppose that K is a field of characteristic 11 > () and f E K[[:l: I, . - . , X n , yJ] has an expre:ssiou r 1'-1 + ... + ar· f ·-y+alY

with ai E K[[:l:), ... , :2:11.1] and p = 0 or p f r. The T.{jchirnhausen transformation of .f is the change or variables replacin~ y with ,

'y =

I

1. hen

al

y+-. 1"

has an expression

f

"='

(y'r

+ b2(y'r- 2 + ... + br

with bi E K[[:r:l,. --, x n ]] for all i.

Exercise 3.9. Suppose that (R, Tn) is a. complete local ring and L 1 C .R is a field such thaL Rim, is finite and separable over Dj_ Use Hcnsel1s Lemma (Theorem 17, Section 7, Chapt.er VIII [92]) to prove that there exists a cueflident field L2 of R containing L 1 .

25

3.3. Blowing up a. point on a nOll-singular surface

3.3. Blowing up a point on a non-singular surface We define the strict transform, the total tran~forUl, and the multiplicity (or order) l/p (C) analogously to the definition of Section 3.1 (More generally, sec Section 4.1 and Definition A.17).

Lemma 3.10. Suppose that X iM u. non-.'1inguLar sur/ace Over an algeb1'Uically do.'1cd field K.. and C is a curve on X. S'uppose that p E: X and vp ( C) = r. Let 1r : B(p) --+- X be the blow-up oj p, C the strict trans/or"Tn of C, and suppose that q E 7r- 1 (p). Then vq(C) < r, and ij r > 0, the1't'. is ai, most one pO'tnt q E 7r- 1 (p) such that vq(C) = r. Proof. Let / = a be a local equation of C at p. If (x, y) are regular parameters in ()X,Pl we can write

f

=

LaijXiyj

,..

asa series of order r in CJx,p "" K[[x,y]]. Let

L

L

=

aijXiyj

'+]=r .

t

f.

leadin~ form of q E 7r- 1(p), then CJB(p),q

be the If

has re~lar parameter~ (Xl, yt} Sllch that

= XI(YI + 0:), or a; = XtYt, Y = Yl·

x = Xl, Y

This substitution imluees t.he inclusion

K[[x,y]] =

K[[Xl,Yl]]'

OX,p --+- OB(l)),q =

First suppose that (Xl, Yl) are defined by

x= Then a local equation for

n.

XI,

Y = a;l(YL

C at q is /1

/1 = for some series

•

=

+ a).

!r, which ha.~ the expansion 1

L ai.1(Yl + a)j +

XIS)

We finish the proof a:; in Section ;i.l.

0

Ak.

Suppose that C ~ the eurve y2 - x 3 = 0 in The only singular point of C is the origin p. Let 1r : B(p) --+- A} be the blow-up of p, C the ~trict transform of C, and E the exceptional divisor FJ = n- 1 (p).

yn

On Ul = spec(K[~, c B(p) we have coordinates XI, YI with :z: = Y = Yl· A local equation for E on UI is Yl = O. Also, y2 _ :L: 3 = y~(l - X~Y1).

A local equation for 6 on U1 CnEn Ul = 0.

is 1 - X~Yl

·T.IY1,

= 0, which is a unit on E n U1 , so

26

3. Curve Singularitieb'

On U2 = Rpec(K[x, ~]) C B(p) we have coordinates XI l Yl with x = Y = XIY1· A local equation for E on U2 is Xl = O. Also\ y2 _ x 3

A local equationfor C on U2 is Thus (} is non-singular, and

-

Xl,

= XT(Y! - Xl).

yr -

:1;1 =

0, which has order < 1 everywhere.

C~C

isa ('eRolution of singularities. Finally,

r(C,Oc)

=

K[Xbyd/(y~ -

:1:1) =

K[ud = K[~l.

This is the normalization of ]([x, y] /(y'2 - x 3 ) that we eomputerl in Section 2.4. As a further example, comlider the curve 0 with equation y2 - X;1 in A2. The only singular point of C is the origin p. Let rr : B(p) ~ A,.2 be the blow-up of p, E be the exceptional rlivi~or, (} be the strict transform of C. There is only one singular point q on C. Regular parameters at q are (J:~l, Yl), where :/: = :1: 1, Y = Xl y,. C has the local equation 2 _ 'l)\ 1 • '1 -Y

0.

Tn this example vq(C) = IIp(C) = 2, so the multiplicity has not dropped. However, this singularity is ,'esolverl after blowing up q, as we calculated in the previous example. These examples suggest an algorithm to resolve curve singularities. First blow up all singular points. IT the resulting curve is not resolved, blow up the new singular points, and repeat as long as the curve is singula.r. This algorithm euds aft.er a finite number of iterations in a non-sinp;ular curvc. In Sections 3.4 and 3.5 we will prove this for curveR embedded in a Hon-singular surface, first over an algebraically closed field K of eharact.eristic zero, and then over an arbitrary field.

3.4. Resolution of curves embedded in a non-singular surface I In this section we consider curves embedded ill a. nOll-singular ~urface over a.1l algebraically closed field K of characteristic zero, and prove that their sin~ularities can be resolved. The theorem is proved by passing to the completion of t.he local ring of the surface at a singular point of the curvc. This allows us to view a local equation of the curve as a power series in two variahles. Our main invariant is the multiplicity of the curve at a point. We know that this Ulultiplicity

3.4. Resolution of curves embedded in a non-singular slll,face T

27

cannot go up after blowing up, so we must show that it will eventually go down,after enough blowing up.

Theorem 3.11. Suppose that (] is a curve on a non-singular surface X over an algebraically clasen field K of characteristic zero. Then thel'e exists a ."Jf:'-ljuence of blow ups of points A : Y ~ X 81£(:11. that the strict transform

-

C of C on Y is non-sing?J,lur. Proof. Let

l'

~sume that

= IUax(llp(C) I p E C}. If r

r > 1. The set {p

= 1, C is non-singular, so we may

Eel v(p)

=

'r} is a subset of the singular

locus of C, which is a proper dosed subset of the I-dimensional variety C, so it is a finite set. The proof is by induction on

'1'.

We can const.ruct a sequence of projective morphisms ... ~ X n ~

(3.2)

... ~

Xo = X,

where each 7Tn+l X nl1 ~ X n is the blow-up of all points on the strict trausform of C which have multiplicity r on If this sequence is finite, then there is an integer n such that all points on the strict transform C n of C have lllUlt.iplidty < r - 1. By induction on T, we can repeat this process to construct the desired morphism Y ~ X whieh induces a resolution of C.

en

en'

We will assume that the sequence (3.2) is infinite, and derive a contra/'" diction. If it is infinite, then for all n E N there are closed points Pn E Cn which have multiplicity r on C n and such that 7("n+1 (Pn+l) = Pn for all n. Let Rn = VxfI.,Pn for all n. We then have an infinite sequence of completions of quadratic transforms of local rings A

"R(J

~

R1

~ ••• ~

Rn

-----+ ••••

Suppose that (x, y) are regular parameters in R.o = OX,p, and f = 0 is a local equation of C in Ro = K[[x, y]]. f is reduced by Theorem. 3.6. After a linear change of variables in (x, y), we may assume that v(f(O, y)) = 1'. By the Weierstrass preparation theorem,

where u is a unit series. We now "complete the r-th power of f". SeL

_ y

(3.3)

= y+

a1(x) r

.

Then

(:i.4)

yT

+ a1(x)y"-1 + ... + ar(x) = it + ~(x)1f-2 + .. , + br(x)

for some series bi(X). We may thus (3.5)

f

= yr

a.~sume

that

+ a2(x)yT-2 + ... + ar(x).

3. Curve Singularities

'28

Since

f

is reduced, we must have some £ti(x)

(3.6)

n

We have n

# O.

Set

. {mult(ai(X))} = mm . . z

> 1, since v(f) = '1'.

"

OX1,Pl has regular parameters Xl, Yl such that

1. x = 2. x

Xl,

Y=

= Xl Yb

Xl

(Yl

+ 0:)

with

ll:

E

K, or

Y = YI

In case 2, where

f

1=

1+

U2(XIyd

Yf

a r (X1Yl) + ... +-----'-

Yl

is a local equation of the strict transform 0 1 of 0 at Pl. IIp1 ( C 1) = O. TllUl:l case 2 cannot occur. In case 1,

II

is

It

unit, so tha.t

I = x~'fl where

r a2(Xl) ( . )r-2 0. .(:1:1) ) + ... + r ' f I = (.Yl + (}' + 2 Yl + 0: 1

Xl

.f1

= 0 is a local equation at Pl for 01. this C8.'3e cannot oecur.

If Q = 0, and vP1 (Cl)

Xl

1£ a

=1=

0, then vP1 (01)


1, so

= r, then 11 has theform of (3.5), with n decreased

by 1.

<

Thllii for some index i with i < n we have that Pi must have multiplicity r on Ci, a contradiction to the assumption that (3.2) has infinite length.

D The transforma.tion of (3.3) is called a Tschirnhausen transformation (Definition 3.8). The Tschirnhausen transformation finds a (formal) curve of maximal contact. H = VOl) for 0 at p. The Tschirnhausen transformation was introduced as a fum.laInental Iuethod in resolution of singularities by Abhyankar.

Definition 3.12. Suppose that C is a curve on a non-singular surface S " over R field K, a.nd p E C. A non-singular curve H = V(g) c spec( "S,P) is a formal curve of maximal contact for C at p if whenever ,..

Sn

---+

Sn-l

---+ • • • ---+

81

---+

spec( 0 S,P)

is a sequence of blow-ups of points Pi E Si, such that the strict transform Ci of C in Si cbntains Pi with vPi (Ci ) = vp ( C) for i < n, then the strict transform Hi, ofH ill Hi contains Pi.

3.5. Resolution of curves embedded in a non-singular surface II

29

. Exercise 3.13. 1. Resolve the singularities by blowing up: a. x'l, = X 4 + y4, b. xy = X 6 + y6, c. X3 = y2 + x 4 + y'" , d. x'l,y + J.:y3 = X 4 + y4,

e. f.

Xn , y4 _ 2X3 y 2 y2

=

4x5 y

+ X6

-

x7•

2. Suppose that D is an effective divisor on a non-singular surfaee S. That. is, there exist curves Ci on S and numbers r i E N such that ID = Iat ... I(;~. D has simple normal crossings (SNCs) on 8 if for every pES there exist regular parameters (x, y) in 0 S,P such that if f = 0 is a local equation for D at p, then J'= unit xUyb in OS,'P'

Find a sequenee of blow-ups of points making the total transform of y2 - x 3 = 0 a SNOs divisor.

3. Suppose that D is all

effe<~tive

divisor on a non-singular surface S.

Show that there exists a sequence of blow-ups of points

Sn

---+ • • • --+

S

such that the total transform 1T'*(D) is a SNCs divisor.

3.5. Resolution of curves embedded in a non-singular surface II In this section we consider curves embedded in a non-singular surface, defined over all arbitrary field K.

Lemma 3.14. 8'uIJIJ(J.lU;~ that K i.~ a field, S is a non-singular surface over K, C is a curve on Sand pEe is a closed po,tnt. Suppose thaI. 7T" : B = B(p) .~ S is the blow-up of p, 6 is the strict transform of C on Band q Err-1(p) n 6. Then and if then K(p)

= K(q). ,..

Proof. Let R = OS,p, (x, y) be regular parameters in R. Since there is a natural embedding of 1r- 1 (p) in B x s spec( R), it follows that x y q E B Xs spec(R) = spec(R[-]) U spec(R[-D. y x

3. Curve Singularities

30

Without loss ofgenerality, we may assUIIle that q E specCR[~]). Let Kl ~ K(p) bea coefficient field of R, m c R[~] the ideal of q. Since R[~]/(x! y) ~ K, [~], t.here exists an irreducible monic polynomial h(t) E K 1 [t] such that

mR[Y] = (x, h( Y)). x x Let fER be such that f = 0 is a local equation of 0, and let r = vp(C). We have an exparusion in R,

f

L

=

aijXiyj

i+j>r

with (L'ij E K1. Thus f = x r /1, where transfonn C of G in spec(R[~]) and

L ..

il =

h

aU (Y)j :r

t+J=r

ill R[~]. Let n

= lIq (C).

= 0 is a local equation of the strict

+ xO

Then

L

h( Y)n divides aij ( Y)j x .. x t+r=r in K 1 [~], Also,

deg(

L

aijCY)i)

..

~+.1=r

in Kd~] impliesn

X


hnplies deg(h(~)) = 1, so that h = ~ - cr with a E K 1 and (by Lemma 3.5) there exist regular parameters (Xl! yd in m q C OB(p),q such that

< rand vq(C) =

OS,p =

is the natural K 1

"-J

l'

K1[[x,y]] ~ Kd[Xl,Yl]] =

OB(p),q

]«(p) algebra homomorphism

x=

Xl,Y

= Xl(Yl

+ a). o

TheoreIll 3.15. Suppose that C is a curve which is a subvariety of a nonsingular surface X over an infinite field K. Then there exists a sequence of blow-ups of points A: Y ~ X .'ru.ch that the .'1trict tran.'1form C of C on Y is non-singular.

Proof. Let r = max{ v p ( C) I p E C}, If r = 1, C is non-singular, so we may assume that l' > 1. The set {p E elI/pC C) = r} is a subset of the singular locus of C, which is a proper closed subset of the I-dimensional variety C,

so it is a finite set, The proof is by induction on r.

3.5. Resolution of curves embedded in a non-singular surface II

31

We can construct a sequence of projective IIlorphisIIIs

(3.7) where each '7rn +l : X n +1 ---7 X n is the blow-up of all points on the strict transform of C which have rnultiplicity r" on On.' If this sequence is finite, then there is an integer n such that all pointl:l on the storiel. transform Crt or C have nlultiplicity < r - 1. By induction on r, we can then repeat this process to conl:itruct the uesired morphism Y ~ X which induces a resolution of O.

en

We will assume that the sequence (:t 7) is infinite, and derive a contradiction. If it is infinite? then for all n E N there are dosed points Pn E On which have multiplicity r on On and are such that 1rn+l(Pn+l) = Pn. Let Rn = CJXn,Pn for all n. We then have an infinite sequence of cmnpletions of quadratie Lransfoflus (blow-ups of maximal ideals) of local rings ~

R

= H{J ~ R 1 ---4

• • • ~.~ R'71,~-~

... .

We will define such that

(3.8) for all i > 1. We can tlms conclude that. Pi, has multiplicity < r on the strict transform Ci of C for some natural number i < 8p + 1. FroHl this contradiction it will follow that (a.7) is a sequence of finite length. ~

Suppose that f E: Ro = CJx,p is such that f =. 0 is a local equation of C ... and (:1:, y) are regular parameters in R = OX,1' such that r =. mult(f(O, y»). We will call such (x, y) good para.meter:; for f. Let K' he a coefficient field of R. There is an expansion

f = ~

aijXiyi

i+j>r

with

(3.9)

aij E

K' for all

i,.i

and

aOr

6(f; x, y) = luin {

We have 8(f; x,y) expression (with 8

# O. Define i . Ij

r-J

< rand

Q.'ij

i=

o} .

> 1 since (x, y) are good parmneterl=i We thus have an

= 8(f;x,y))

j' =

~ i+jo>r6

n.'ijXiyi

= Lo +

~ i+j6>ro

ai:ixiyj,

•

32

3. Curve Singularities

where

(3.10)

E

La =

a'ij:l:iyi

=

a.OryT

+A

i+j5=ro

=F 0 and A is not zero. Suppose t.hat (Xl y) are fixed good parameters of i. Define (3.11)

is such that

aO r

n

8'P

= sup{o(/; x, YI) ly =

Yl+

E bix

i

with

'n

EN and b'iE K'} E ,:rNU{oo}.

i=l

"\\I"c cannot have

op = 00,

since then there would exist a serie.<. t 00

+ Ebi Xi

Y = Yl

i=l

such that 0(/; x, YI) = that

00,

and thus there would be a unit series '"'( in R such

f

=

,YI'

But then ',. = 1 since f is reduced in R~ a contradiction. We see then that op E ~N. After possibly making a substitution n

Y = Yl

+ Ebix i i=l

with bi E ](', we may a8811Iue that 8}) = (5(1 j X, Y)' Lct 6 =

op.

Since vP1 (C\) -=

'T',

by Lemma. :t14, we have an expression

R1 = K'Uxl, YI]l, where R ~ Rl is the natural K' -algebra homomorphism such that either X=Xl,

for some

0'

E

K',

Y=XI(Yl+O')

OJ'

We first consider the case where a: C 1 (in spec(Rl» is It = 0, where

= Xl Vl, Y = Yt,

!I = r1 = '" ~ aijxi. +YIO = Yl

for sonle

n~ g,

aUT

A local equation of

+ Xlg + YIlt

i+J:::r

h E RI. Thus

il

is a unit in R 1 , a contradiction.

3.5. Resolution of curves embedded in a non-singular surface II

Now consider the case where:r.: = w1, Y = A local equation of 0 1 is

L=~ ~

J1 = xr

for some

n E R1 .

Bince

V,l1

(01)

L

(YI

+ a)

with 0

:I

0

E 1('.

ai'(Yl +a); +x10 J

i+.i=r

1

Xl

33

= r~

aij (YI

+ Q)j = an"Yl'

i+j=r

Substituting t

= Yl + 0, we have

L

aijtj = aor(t -

or·

i+j=1'

If we now substitute t leading forrn L of f,

L

'J =

=

~ and rnttltiply the series by x r , we obtain the

ai.ixiy:i -' aor(Y - axr = UOryr

+.,. + (-l)fao r a r x r .

i+.i-=r

Comparing with (3.10), w€ see that r = r'd and 8 =- 1, so that Ltj = L. But we can replace y wit.h y-n:r. to increase 6, a contradiction to the maximality of 6. Thus lIPl (el) < r, a contradiction. Finally, consider the case x equation of C1 ilS J1

=. Xl, Y

~ a' 'xi+j-r J LJ tJ 1 Yl

= Lxi --

A

L

i+jl5~rJ

a:Z-i +1., j x

lY{ +

i+j6'=1'(S'

where I

= i +j

L

ai-i+r.jxiYi,

i+jlj'>rD'

- r. Since

IJo'(xl, Yl)

1

= rLo(XI, XIYI) ~rl

has a.t lea..,t two non-zero terms, we see that 6 (II; We will now 15how t.hat 8P1 make a substitution

Yl with 0

=1=

Set 8' = 8 - 1. A local

= Xl Yl.

= Yl, -

= t5(ft; x}, yd.

Xl,

Y.) = 6' = t5 - l.

Suppose not. Then we can

"'b l + l 'l1gJer } . L.- iXIi = Yl' b - lXl order terms ill

Xl

b E K' such that

6(ft;xl,yi) > t5(ft; Xl, yt}. Then we have an expression ~ L.J £+j6'=r6'

a-i-j+r,j X,.il (y'1

-

bXld)j

--

aO r (y'1 )r

+

~ L.J i+jD'>r6'

b-~J-xi'1 ( Y1')j 1

3. Clll've Singularities

34

so that 6'

= dE N,

8Jld

~ LJ

ai_j+r,jxli (111' - bXld)i =

lto-,. (Yl') r

.

i+jo'=ro' Thus ""' C:tif];l . i+i-'I'~j" LJ .!It = ~ a t _j +r,j:l:li"J bJ = i+j6=ro i+.i6'=r6

(

aOr Yl

+ bxlo')r .

1

If we now mnltiply these series by

Lo -."

E

xI,

we obtain

ai.jxiyi = aor(Y + bxJr·

i+io='r& But we can now make the substitution y'

= 1/ -

bx 6 and see that

de!; :r" y') > 6(1; x, y) = 6p , a contradiction, from whkh we conclude that 6P1 = 8' = 6p - 1. We can then illduetively define 0p. for i > 0 by this procedure so that (3.8) hold~.

The conclusions of the theorem now follow.

0

Remark 3.16. 1. This proof is a genera1i7.ation of the algorithm of Sedion 2.1. A more general version of this, valid in arbitrary two-dimensional regular loeal rings, can be found in [5] or [73}.

f

which achieve 6p = 6(j; x, y) are SllCh that y = 0 is a (formal) curve of maximal contact for C at p (Definition 3.12).

2. Good parameters (x, y) for

Exercise 3.17.

op defined ill formula (3.11) is equal to &p = sup{d'(j; x, y) I (x, y) are good parameters for fl·

1. Prove that the

ThULe:; 8p does not depend 011 t.he initial choice of good parameters, and 5p is an invariant of p.

t

2. Suppose that vp(C) > 1. Let·1r : B(p) ~ X be the blow-up of p, be the strict tram;forll1 of C. Show that there is at most one point q E 1T- 1(P) such that lIq(C) = vp(G). 3. The Newton polygon N (j j X, y) is defined as follows. Let I be the ideal in R = K[[x y]] generated by t.he monomials x Ct y/3 such that the coeffieient a u l3 of xOyfl in lex, y) = 2: lt Q {3a;<.r y13 is not zero. Set 1

P(f;x,1/) = {(a,t/) E Z21xO:y# E I}. Now define N(f; x, y) to be the 8Inallest convex subset of R2 sueh that N(j; x, y) contains P(f; x, y) and if (O!, [J) E N(fj x, y), (8, t) E

3.5. Resolution of curves embedded in a non-singular surface II R~"

then (0:

+ 8 , j3 + t)

E N{fj x, y).

35

Now suppose that (x, y)

are good parameters for I (so that v(f(O, y» = v(l) = r). Then (0, r) E N(f; X, y). Let the slope of the steepest segment of N(lj x, y) be 8(lj:£, y). We have 0 > s(f; x,y) > -I, since a.'-~t3 = 0 if a + 13 < r. Show that

o(!; x, y)

1

= - 8( fjX,y )'

..........

Chapter 4

Resolution Type Theorems

4.1. Blow-ups of ideals Suppose that X is a variety, and :J c

(9 x.

is an ideal sheaf. The blow-up of

:J is 1r :

B = B(:I) = proj(EB J'l) ~ X. n>O

B is a variety and 1r is proper. If X is projective then B is projective. 7T is an isoIllorphisIIl over X - V(:I), and :rOB is a locally principal ideal sheaf.

c X is an open affine subset, and R (11, ... ,1m) c R, then If U

7r-

l

= r(U, Ox), I = r(U,:I) =

(U) = B(l) = proj(EB P). n>O

If X is an integral scheme, we have m

proj(EBr~) = U8pec(R[~, ... , I~t]). n>O i=l j,,, It Suppose that W c X is a subscheme with ideal sheaf Iw on X. The total transform of W, 7r*(W), is the subscheme of B with the ideal sheaf

I1l'-(W)

= IWOB·

Set-theoretically, 1r*(W) is the preimage of W, 1r- 1 (W).

37

38

4. Resolution Type Tlleorems

Let U = X - V (3). The strict transform of 7r- 1 (W n U) in B(,J).

TV of W

is the Zariski clOSIlI'e

Lemma 4.1. Suppose th
U(I : Jft) =

n

qi·

{il.Ttpd

n=O

The proof of Lemma. 4.1 follows easily from the definition of a primary ideal. Geometrically, Lemma 4.1 says that U~ 0(1 : In) removes the primary eomponents qi of 1 such that V(qd c V(J). ""

Thus we see that the strict transform W of W has the ideal sheaf 00

I

n = U(IwOn: 3 OB). w n=O

For qE B,

IW,q =

{f

E OB,q

If ,J;t c IWOB,q

for some n

> O}.

The strict transform has the property that

W=

B(30w) - projCEB(,-1 0 w

r l

)

n>O

This is shown in Corollary II. 7.15 [47].

Theorem 4.2 (Universal Property of Blowing Up). Suppose that I is an -i.deal sheaf on a variety V and f : W --.. V is a morphism of varieties such that IOw is locally principal. Then there is a unique morphism 9 : W ~ B(I) such that f = 1t" 0 g. This is proved in Proposition II.7.14 [47].

Theorem 4.3. Suppose that C irS an integral curoc over a field K. Consider th.e sequence ... --+ Cn ~ •.. --..

(4.1)

C'!

~ 0,

where C n + 1 --+ Cn is obtaine4 by blowing up the (finitely many) singular points on C n . Then this sequence is finite. That is, there exists n such that Cn is non-,'ting1LLar. of generality, C = spcc(R) is affine. Let R be the integral closure of R in the function field K (C) of C. R is a regular ring. Let C ~ spec(R). All ideals in R are locally principal (Proposit.ion 9.2, page

Proof. Without

1088

94 [13]). By TheoreIu 4.2, we have a factorization C

~

en ~ ... --.. C1 --+ C

39

4.1. Blow-ups of ideals

for all n. Since C -+ C is finite, Ci +1 --+ Ci is finite for all i, and there exist affine rings Ri such that Ci = spec(Ri). For all n we have sequences of inc1 usions

R -+ R 1

-+ ... -+

Rn -+ R.

Here Ri 1= Ri 11 for all i, since a maximal ideal m in 14 is locally principal if and only if (Rdm is a regular local ring. Since R is finite over R, we have that (4.1) is finite. 0

Corollary 4.4. Suppose that X is a variety over a field K and C is an integral curve on X. Con!;;ider the sequence . . . ) X n '!'.~ X n -1

o}

•••

~

X1 ~ X,

where 7ri+l is the blow-up of all points of J""<. which are singular on the strict transform Ci of C on Xi. Then this sequence is finite. That is, there exists an n 8'uc:h that the. 8trid. t1'UnsJU1'1n Cn of C is non-singula1·.

Proof. The induced sequence .. ' - 1

C71.

--+... _..~

01

--)

C

of blow-ups of points on the strict transform Ci of C is finite by Theorem 4.3. D TheorerIl 4.5. Suppose. that V and W m'e varieties and V -+ W is a projective birational morphism. Then V '" B(I) for some ideal sheaf I C Ow.

This is proved in Proposition II. 7.17 [47]. Suppose that Y is a non-singular suhvariety of a variety X. The monoidal transform of X with center Y is 7r : B(Iy) -+ X. We define the exceptional divisor of the monoidal transform 1r (with center Y) to be the reduced divisor with the same support EL"l 7t""'(Y). In general, we will define the exceptional locus of a proper birational morpism ¢J : V ~ W to be the (reduced) closed subset of V on which cjJ is not an isomorphism.

Remark 4.6. If Y has codimension greater than or equal to 2 in X, then the exceptional divisor of the monoidal transform 7r : B(Iy) ~ X with center Y, is the exceptional locus of 7r. However, if X is singular and not locally factorial, it may be possible to blow up a non-singular codimellsion 1 subvariety Y or X to get. a. Illolloidal transform 7r such that the exceptional locus of 7r is a proper closed subset of the exceptional divisor of 7r. For an example of this kind, see Exercise 4.7 at the end of this section. If X is non-singular and Y has codimension 1 in X, then 7r is an isomorphism and

4. ResolutioIl Type Theoren18

40

we ha.ve defined the exceptional divisor to be Y. This property will be useful in our general proof of re~olution in Chapter 6.

Exercise 4.1. 1. Let K be an algebraically doseu field, and X be the affine surface

Let Y = V (x 7 z) C X J and let 7r : B ---+ X be the monoidal transform centered at the non-singular curve Y. Show that 1r is a resolution of singularities. Compute the exceptionalloeus of 1r, and cOIupute the exceptional divisor of the monoidal transform 7r of Y. 2. Let K be an a.l~ebraically closed field, and X be the affine 3-fold

x = spec(K[x, y, z, w]j(xy -

zw).

Show that the mOllodial transform 7r : B ~ X with center Y is a resolution of singularities. in each of the following cases, and describe the exceptional locus and exceptional divisor (of the monoidal transform) : a. Y = V( (:r, y, z, b. Y=V((X,Z», c. y = V«y, Show that the monoidal transforms of cases band c factor the ruonoidal trarustoflll of case a.

w»,

z».

4.2. Resolution type theoreIllS and corollaries Resolution of singularities. Suppose that V is a variety. A resolution of singularities of V is a proper birational morphism ¢ : W ---+ V fSuch that W is non-singular.

Principalization of ideals. Suppose that V is a non-singular variety, and I C Ov is arl ideal sheaf. A principali:lation of I is a proper bh'ational morphism cP : W ---+ V such that W is non-singular and IOw is locally principal. Suppose that. X is a non-singular variety~ and I c Ox is a locally principal ideal. We say that I has simple nonnal crossings (SNCs) at p E X if there exist regular parameters (:r.I, ... , x n ) in OX,p such that. I p X~l ... X~'·OX,P for some naturalllumbers all." 1 an'

4.2. Resolution type theorems and corollaries

41

Suppose that D is an effective divisor on X. That is, D = 'nE1 + ... + rnEn , where E i are irreducible codimension 1 subvarities of X, and ri are natural numbers. D has SNCs if ID = Ii"1 .. ·l1nn. has SNCs. Suppose that W C X is a subscheme, and D is a divisor on X. We say that W has SNCs with D at p if

1. W is non-singular at p,

2. D is a SNC rlivilmr at p,

3. there exist regular parattleters (Xl, ... ,:1:n ) in OX,p a.nd r < n such that Iw,p = (Xl,_ .. ,xr ) (or Iw,p = Ox,p) and I for some

ai

E

D,p

= :1.'1.a1 ... of'n

,."an Pl

VX,p

N.

Embedded resolution. Suppose that W is a subvariety of a non-singular variety X. An embedded resolution of W is a proper morphism 1r : Y -+ X which is a product of monoidal transforms, such that 7T is an isonlorphism on an open set which intersects every component of W properly, the exceptional divisor of 7T is a SNC divisor D, and the strict transform W of W has SNCs with D.

Resolution of indeterminacy. Suppose that 4> : W -+ V is a rational map of proper K-varieties such that W is non-singular. A resolution of indeterminacy of 4> is a proper non-singular K -variety X with a birational morphism 1/J : X ~ W and a morphism ..\ : X -+ V such that A = 1> 0 '1/).

Lemma 4.8. SUl)lJO."e that resolution of singularities is true for K -varieties of dimension n. Then resolution of indete17ninacy iB tnLf~ for rational maps from K -varieties of dimension n.

Proof. Let 4> : W

-+

V be a rational ma.p oj' proper K-varieties, where W

is non-singular. Let U be a dense open set of W on which 4> is a Inorphisrn. Let r (P)). By resolution of singularities, there is a proper birational lllorphism X~ r q, such that X is non-sin~ular. D

Theorem 4.9. Suppose. that K is a perfect field, resolution of singularities is true for pro.iective hypersurfaces over K of dimension Tl., a:n.d principalization of ideal.9 i.9 true for non-singular varieties of dimension n over K. Then resolution of singularities is true for projective K -varieties of dimension n.

,4. ResulutiOIl Type Theorems

42

Proof. Suppose V is an n-dimensional proj~ctiveK-variety. If Vl, .. . , v;. are the irreducible components of V, we have a projective hirational morphism from the disjoint union of the Vi to V. Thus it suffices to assume that V is irreducible. The function field K(V) of V ~ a finite separable extension of a rational function Held K(Xl,' .. ,xn ) (Chapter II, Theorem 30, page 104 [92]). By the theorem of the primitive element,

K(V)

"-J

K(Xl,' .. , x n )[xn +l]/(f).

We can clear the denominator of

f

so that

, i"+l L-Ja.·i, ... in.+IXti l •. 'x n +1 f =~

is irreducible in Krxl, ... ,:l:n+l]. Let

d = max{h

+... i n+l I ail ... in+l =1= O}.

Set

. . Xd-(il+ .. ·+in+l)xil .. . Xin+l F -- ~ L-J (ttl ... ?'n+l 0 l ' n+1 , the homogenization of f. Let V be the variety defined by F = 0 in pn+l. Then K(V) C:t K(V) implies there is a birational rational map from V to V. That is, there is a birational morphiRlll eP : V ---+ V, where if is a dense open subset of V. Let r ¢ be the Zariski clusure of {(a, (p(a)) I a E V} in V x" V. We have birational projection lnorphisms 1r1 : r -+ V and 7f2 : r ¢ ---t V. f¢ is the blow-up of an ideal sheaf ..1 on V (Theorem 4.5). By resolution of singularities for n-dimensional hypersurfaces, we have a resolution of singularities f : W' ~ V. By prilldpali~ationof ideals in nonsingular varieties of dinlcnsion n, we have a principalization 9 : W ~ W' for .JOW" By the universal property of blowing up (Theorem 4.2), we have a morphism h : W ~ r (j>' Hence 7rl 0 h : W ---t- V is fI. resolution of singularities. D

Corollary 4.10. Suppose that C is a projective C'UTwue

over

an infinite perfect

field K. Then (] has a rClwl1Ltion of singularitieB.

Proof. By Theorem 3.15, resolution of singularities is true for projective plane curves over K. All ideal sheaves on a non-singular curve arc locally principal, since the local rings of points are Dedekind local rings.

D

Note t.hat TheoreU14.3 and Corollary 4.4 are stronger results than Carol. lary 4.10. However, the ideas of the proofs of resolution of plane curves in Theorems 3.11 and 3.15 extend to higher dirneusions, while Theorem 4.3 does not. As a consequence of embedded resolution of curve sinbrularities on a surface, we will prove principali~at,ion of ideals in non-singular varieties of dimension two. This simple proof is by Abhyankar (Proposition 6, [73]).

4.2. ResoluUoll type theorems and corollaries

Theorem 4.11. 8uppo.'1C that 8 is a non-singular surface over a field K, and ..1 C Os is an ideal sheaf. Then. ther'e exists a finite sequence oj blow-ups a/points T -+ S such that .:JOT is locally principal. Proof. Without. loss of generality, 8 is affine. Let

J

= f(..1, Os) = (11, ... ,1m).

By embedded resolut.ion of eurve singularities on a surface (Exercise 3.13 of Section 3.4) applied to vlt _. -1m E 8, there exists a sequence of blow-ups of points 7rl ~ 8,-) S such that b ... Jm = 0 is a SNC divisor everywhere on 81. Let {PI, ... , Pr} be the finitely many points on 81 where J 0 S1 is not loeally prindpal (they are contained in the singlll81' points of V(y'JCJs t )). Suppose that pE {PI, ... , Pr }. By induction on the number of generators of .I, we may assume that J081,P = (f,g). After possibly multiplying f and 9 by units in OSl,P' there are regular paramet.ers (:l:,y) in (')Sl,P such t.lJ3,t

f = XUyu,

9

= xCyd.

Set lp = ((1, - c)(b - d). (1,9) is principal if and only if t p > O. By our assumption, t p < O. Let 11"2 : 8'2 -+ 8 1 be the blow-up of p, and ~mppose that q E 7r-' (p). We will show that t q > t p • The only t.wo points q wher'p, (!, g) may not. he principal have regular parameters (Xl, Yl) such that :1: = :r.l, Y = :J;lYl or x

If x

= Xl, Y =

= Xl Y' ,'Y = y, .

XIYl, then

g_xr.+dydl' f --xa+hyO 1 1'- 1 (l' - d) 2 + (0. - c) (b - d) > tpThe ease when :1: = :tlYl, Y = Ylis simila.r. By induction on min{tp I JOtit,P is not principal} we can principalize J after a finite nnmber of blow-ups of points. D tq

= (it + b - (c + tI)) (b - d)

=-

~

Chapter 5

Surface Singularities

5.1. Resolution of surface singularities In this section, we give a simple proof of resolution of surface singularities in characteristic zero. The proof is by the good point algorithnl of Abhyankar

([11, [62], [73]). Theoreul 5.1. Suppose that S is a projective surface over' CUt u,lgeb'raically closed field !{ of charader'istic O. Thfm there exists a resolution of singula""'itic,~

A:T

----+

S,

Theurem 5.1 is a consequence of Theorems 5.2, 4.9 and 4,11.

Theorem 5.2. 8'uIJpmic that .9 is a hypersurface of dimension 2 in (l, 'f/.U17,singular variety V of dimension 3, over an algebraically closed field ]( of cha1'acteristic 0, Then there exists a sequence of blow-up.Ii of points and non-singular' (:ur'Ve~ conta'ined in the strict transform Si of 5,

such that the str-iel

lTYL71u 9form

8'11. of S on Vn is non-sin,qular.

The remainder of this sect.ion will be devoted to the proof of Theorem 5.2. Suppose that V is a non-singular three-dimensional variety over an algebraically c1osp,d field K of characteristic 0, and S c V is a surface. For a natural number t, define (Definitions A.17 and A.20) Singt(S)

= {p E V I vp(S)

~

t}.

By Theorem A.19, Sing t (S) is Zariski elosed in V. 45

5. Surface Singularities

46

Let r' .,.... max{ I,

I Singt(b') f

0}

be the rnaxirnal llmltiplicity of points of S. There are two types of blow-ups

of non-singular subvarieties on a non-singular three-dimensional variety, a blow-up of a point, and a blow-up of a non-singular curVf~. We will first consider the blow-up 1T' : B(P) --+ V of a closed point p E V. Suppose that U = spec{R) c V is an affine open neighborhood of p, and p has ideal u"p =- (~l;, y: z) C R. Then 7[-1 (lJ)

= proj(ffi 1nnp ) = W

spec(R[Y, Z]) U spec(R[X, Z]) U spec(R[X. Y]). :l: ~l;Y y z z

The except.ional divisor is E =

7r-- 1 (p)

~ [p2.

At each closed point q E 7t'-1(pL we have regular parameters (X1,Yl,Zt} of the following forms:

x = with

Ct,

{3 E !(, Xl

Zl

0'

E K, Yl

= Xl (Yl + a),

z=

Xl (Zl

I

fJ),

= 0 a local equation of E, or X

with

Y

Xl,

= XIYl,

Y = Yl,

Z

= Yl(Zl + Q)

= 0 a local equation of E, or

= 0 a local equation of E.

We will now consider t.he blow-up 7r : B(C) --+ V of a non-singular curve C C V. If p E V and U = spec( R) c V is an open affine neighborhood of p in V such that 'm p = (x, y, z) and the ideal of (J is 1 = (x, y) in R, then 7r- 1 (U)

= proj(ffi In) = spec(R[x]) W y

u spec(R[Y]). :r

Also, 1r- 1 (p) ,...., !pI and 7[-1 (C n U) ,...; (C n U) X pl. Let E = 7r- J (C) be t.he exceptional divisor. E is a projective bundle over C. At each point q E 1r- l (p) I we have reg;ular parameters (Xl, Yl , Zl) such that x

where

0' E

= Xl,

Yl =

Xl(Yl

+ aL

z = Zl,

K, :z: 1= 0 is a local equation of E) or :1:

where

Y=

= X1Yb

Y

= Yb

Z = Zb

0 is a local equation of E.

In this section, we will analyze the blow-up 7r:

B(W)

= B(Iw)

~

V

of a non-singular subvariety W of V above a closed point 1) E V, by passing to a formal neighborhood spec(Ov,p) of p and analyzing the map 7t' : B(Iw,p) --+ spec(Ov,p). We have a natural isomorphism B(Iw,p) r-v

47

5.1. Resolution of surface singularities

B(Iw) 7f -1

(1J)

spec(Oy,p). wit h 7r'" 1 (1)). Xy

Observe that we have a natural identifica.tion of A

We say that an ideal 1 C OV,p is algebraic if there exists an ideal J c OV,l' such that I = J. This is equivalent to the statement that there exists " au ideal sheaf Ie Ov such that I p = f. A

A

IT I C C'Jv,p is algebraic, so that there exists an ideal sheaf I c Oy such " that Ip = I, then we can extend the blow-up 1r : B(I) -+ spec(Ov,lJ ) to a blow-up 7r : B(T) ~ V. ~

n

The maximal ideal mpOy is always algehraic. However, the ideal shear I of a non-singular (formal) curve in SPPA:(Ov,p) may noL be a,lgebraie.

One example of a formal, non-algebraic curve is

6 Ai ,0'

I = (y - eX) C C[ [x, y]] =

A more subtle exa.mple, which could occur in the course of this section, is the ideal sheaf of the irreducible curve

J = (y'l - x2

-

x3 ,z) C R == K[x,y,z], A

which we studied after Theorem 3.6. In R parameters

x = y - xv! + x,

= K[[x, y, zJJ we have regular

y = y + xvI

+ x,

Z -.." N.

Thus

JR = (xy, z) = (x, z) n (y, z) c R = K[[x, y, z]]. In this example, we may be ternpted to blow-up one of the two formal branches x = 0, z = or y = 0, z = 0, but the resulting blown-up scheme will not extend to a blow-up of an ideal sheaf in

°

Ak'

A situat.ion which will aril=ie in this section when we will blow up a formal curve which will actually be algebraic is given in the following lemma. Lemma 5.3. Suppose that V is a non-sin.qular three-dimensional variety, p E V is a closed point and 7r : Vi = B(p) ---4 V i,'I th~ blow-np of p with e:r.ceptional d'i71i,~or E = 7r l(p) ~p2. Let R = VV,p. We have a commutative diagmm of morphisms of schemes

B

= B(mpR)

-;

such that

that I cxist~

c

A

7r- 1 (mp ) --+ 7r- 1 (p)

_

= B(P)

Lrr

7f!

spec(R)

Vi

--+

V

= E is an isomorphism of schemes. Suppose

R is any ideal, and I C On is the strict transf0'f'1T!: of 1. an ideal ,.,heaf:l on VI .'3'uch that ..10B = I EOB + I.

rph~n

there

5. Surface SiI}guJa.rities

4H

Proof. JOE is an ideal sheaf on.. E, so there exists an ideal sheaf:r c OVI such that IE C .:r and :fILE ~ lVE. Thus :r has, the desired property. 0 Lemma 5.4. Suppose that V is a non-.'ling'lllar three-dimensional variety, [; c V is a surface, C c Singr(S) is a non-singular curve, 7r : B(O) -+ V is the blow-up of C, and ,§ is the .'tI.1"idtran.'1form of S in B(C). Suppose that IJ E C. rphen l/q(S) < r for all q E 1T- 1(p), and there exists at most one point q E 1f- 1 (p) such that vq(S) = r. In particular, if E = 1r- 1 (C), then either Singr(~) n E is a non-singular curve which maps iSOTnorphi(:(Llly onto C, or Sing.,.(S) n Eisa finit.e union of lwint'l. Proof. By the Weierstrass preparation theorem and after a Tschirnhauscn transformation (Definition 3.8), a local equation f = 0 of S in Vs,p = K[[x, y, z]] has the I"onn

f

(5.1)

(r) ... _ "

= zr .

+ a2(x,Y)Zr 2 + ... + ar(x,y). · af

...

1

_

XC ,P ,

ar ... '--1

."

and r!z - w=r E Ie,p' 'rh us " z E Tc,p. Aft.er a change of variables in x and y, we may assume that Ic,p = (x, z). Then f E irnplies x·i. I (J,i for all i. If q E 1T- I (p), then But

f

t:

... OB(C),q

(Ic,p) - 'Ib,p ImplIes liZ

E

tc,p

has re~ular parameters (:r:I, y, Zl) ~mdl that

x

= Xl Zt,

Z = Zl

or

for some n E K. In the first case, a local equat.ion of t.he strict transform of S is a unit. In the second case, the strict transform of S has a local equa.tion lJ.2 ( )1' 2 ar Zl + n - + ... + -:;:. f 1 = (Zl + Q ). + ~ 1'

Xl

We have v(ft)

< r, and 1/ (11) < r if Q

Xl

=1=

o.

0

Lenlma 5.5, Suppose that p E Sing,.(S) is a point,

B(p) -lo V is the blow-up of p, S is the strict transf017n of [3 'in B(p), and E = 7r- 1 (P). Then T/q(S) < T' for'cdl q E 7r- 1 {]J), and eitherSingr(S)nE isa non-singular curve or' SiTlgr(S) n E i.'1 a finite union 0/ points. 1T :

Proof. By the Weierstrass preparation theorem and after a Tschirnhausen transformation, a local equation f = 0 of /3 in OS"]J = K[[x, y, z]] has the form

(5.2)

f = z'" + a:.! (x, y)zr-2 + ... + lt ·(:1:, y). 1

49

5.1. Resolution of surface singularities

If q E 1T- 1 (p) and lIq(S) > such that

T,

then

OR(C),q

has regular parameters

(XI,

y, Zl)

or :I;=:l:1~

y=a:l(YJ+n),

Z=X1 Z ,

nE

i~ contained in the line which is the intersection of E with the strict transform of z = 0 in H(O) Xv spec(OV;p). D

for some

0'

E

K, and Vi(S) = t. Thus Singr(S)

Definition 5.6. Sing.,.(S) has simple normal crossings (SNCs) if 1. all irreducible components of Singr(S) (which could be points or curves) are non-singular, and

2. if p is a singular point of Singr(S), then there exist regular parameters (a:: Y, z) in OV,p such that ISingr(S),p = (xy, z). Lemma 5.7. Suppose that Sing 7• (S) has simple normal crossings, tV its a lJO'int or o,ni1"1'f'Aucible curve in Singr (S), 1f : V' = B (W) ---t V is the blowup of Wand S' is the strict transfonn of 6' on V'. Then Singr (8') has simple normal crossings. Proof. This follows frOIlI Leulluas 5.4 and 5.5 and a. sirnple local calculation on V'. n Definition 5.8. A closed point p E 8 is a p1'cgood point if, in a neighborhood of p, Singr.(S) is either empty, a non-singular curve through p, or fI, a union of two non-singular curves intersecting transversally at p (satisfying 2 of Definit.ion 5.6).

Definition 5.9. pES is a good point if p is pregood and for any sequence

Xu

~

X n-l

---4 •••.~

Xl-~

spec(Ov,p)

of blow-ups of non-singular curves in Sing,.(Si), where 8" is the st.riet transform of ..9 n spec(Ov,p) on Xi, then q is pregood for all closed points q E Sing,.(Sn)' In particular, Singr (8n ) eontains no isolated points.

A point which is not good is called bad. Lemma 5.10. Suppose. that all points of Singr (8) lLre good. Then there exists a sequence of blow ups V' -, \!;l,

---t ••• ---t

VJ

~ V

of non-singular curves contained in SinP;7.(Si), where Si is the stdct t1U1tSform of 8 in \Ii, such that Singr(S') = 0 if S' is the strict transform of S on V'.

50

5. Surface Singularities

Proof. Suppose that C is a non-singular curve in Sing r (S). Let 7T"1 : Vi = B(C) ~ V be the blow-up of C, and let 5\ be the strict transform or s. If Singr (8t} =F 0, we can choose another non-singular curve Ct in Singr(Sl) and blow-up hy 7r2 : V2 = B(Od ----+ V1. Let..'h be the strict transform of 8 1. We either reach a surface Snsuch that Singl'(Sn) = 0, or we obLain an infinite sequence of blow-ups ...

~

Vn

~

\l,t- 1

----+ • • • ~

V

such that each Vi+l ~ Vi is the blow-up of a curve Ci in Singr(Si), where Si is the strict transform of 8 on Vi. Each curve OJ which is blown up must map 0111,0 a. ClIl've in 8 hy Lemma 5.f), Thus there exists a curve I C S such that there are infinitely many blow-ups of curves ma.pping 011 to , in the above sequence. Let R = OV,'"Y a two-dimensional regular local ring. Is,'Y is a height 1 prime ideal in this ring, and P = 'L"Y,"Y is the maximal ideal of R. \Ve have dimR + trdegK RJP = 3 1

by the dirnerusion formula (TIH:lorern 15.6 [66]). Thus R.JP haR transcendence degree lover !(. Let t E R be the lift of a transcendence basi~ of RJP over K. TI1fHl K [t] n P = (0), so the field K(t) c R. We can write R = AQ~ where A is a finitely generat.edK(t,}-a.lgebl'a. (whieh is a doma.in) and Q is a maximal ideal in A. Thus R is the local ring of a non-singular point q on the K (t) surface spec(A). q is a point of nlultiplicity r on the curve in spec(A) with ideal sheaf Is,"Y in R. The sequence

...

~

Vn.

Xv

spcc(R)->

Vn-l

Xv spec(R)

---+ •.. --

spec(R)

consists of infinitely many blow-ups of point.s on fL K(t)-surface, which are of multiplicity r on the strict transform of the curve spec( RJIs,'Y ). But this is impossible by Theorem 3.15. D

Lemma 5.11. The nurnber of bc£d points on 8 'is finite..

Proof. Let Bo - {isolat.ed points of Singr (8)}

U {singular points of Singr(S)}.

Singr(S) - B u is a non-singular I-dimensional subseheme of V. Let. 1rl :

Vi = B(Sing.r (8) - Bo) ~ V - Eo

be its blow-up. Let 8 1 be the strict transform of S, and let

B1

=

{isolated points of Singr (5\)} U {singula.r point.s of Singr (81)}'

We can iterate to construct a sequence . •.

-t

(Vn

-

Bn )

---t •••

---+

V,

51

5.1. Resolution of surface singularities

where 'lrn :

~t

-

~

Bn

V - Tn are the induced surjective maps, with

Bo U ?r1(BI ) U ., , U 1l"n(Bn ).

Tn =

Let Sn be the strict. t.ransform of S on Vn . let C C Sing1.(B) be a curve. spec( OS,G) is a curve singularity of IJlUItiplicity r , embedded in a non-singular 81ll'fac:e over a field of transcendence degree lover K (as in the proof of Lemma 5.10). ?rn induces by base change Sn xsspee(Os,c) ~ spec(Os,c), which corresponds to Illl open subset of a sequence of blow-ups of points over spee(Os,c). So for n » 0, Sn Xs spec(Os,c) is non-singular. Thus there are no curves in Singr(Sn) which dominate C. For 11. » 0 there are thus no curves in Singr(Sn) whieh dormnate curves of Sing7·(B) , so that Sillgr(Sn) n (Vn - Bn ) is empty for large n, and all bad points of 8 are contained in a finite set Tn. 0

Theorem 5.12. Let .. .

(5.3)

._~

lI.n

--+

v:

'1'1.- ·1

---7 . . • ---7

~l ~

V

be the sequence where 1rn : Vn ~ Vll.-1 i.9 the blow-up 0/ all bad points on the strict tran.9form 811.-1 0/ S. Then this sequence is finite, so that it terminates ajte1' a finite number of steps with a Vm such that all points of SingrU'm) are good. We now give the proof of Theorem 5.12. Suppose there is an infinite sequence of the fornl of (5.3). Then there is an infinite sequence of points Pn E:; Sing r (8n ) such that. 'lrn(Pn) = PTI 1 for all n, We then have an infinit.e sequence of homomorphisms of local rings ~

Ro = "V,p

·4

R1 =

-

OVI ,PI

---7 • • •

~

R,t

~

= Ol-~ ,PI1

~ ,,• .

By the Weierstrass preparation theorem and after a. Tshirnhausen transformation (Definition 3.8) there exist regular parameters (Xl Y, z) in that there is a local equation f = 0 for S in Ro of the form

f

=

z"

Ro

such

+ a2(x, Y)ZT 2 + ... + ar(x , Y)

with v (a..i (x, y)) > i for all i. Since IJpi (Bi ) = r for all i, we have regular parameters (Xi, Ui, Zi) in Ri for all i and a local equation h = 0 l'or 8i ::Iuch that Xi-l

with

ai E

= Xi,

Yi-l

= Xi(Yi

I-

Qi),

Zi-l

= Xi-1Z'i-l

K, or Xi-l

= XiYi,

Yi-l = Yi,

Zi-l

= Y'i-1Z'i-l,

and f Ji

( ( = Zir + a2,i Xi, Yi ) Zi'1'-2 + ... ar,iXi, Yi )

I

52

5. Surface Singularities

where

for all j. Thus the sequence

K[[x, y]

--t

K[[~l;] 1 YI]] ~

...

is a sequence of blow-ups of a maximal ideal, followed by completion. In l rno. Thus for i >mo we have an expression

rr;=2

f i = Zir +-a2i ('Xi,Yi )Xia:Z'1 Yiboli,Zir-2 + ... +-ari. (Xi,Yi') Xin'd Yibri , where the ajiare units (or zero) in Rri, and aji + bji ~ .i if aji f; O. o. 4')

( I':

We observe that, by the proof of Theorem A.19 and Remark A.21, ISiug1.(Si),Pi

=

-I'J,

where J is the ideal in Ri generated by fJi+k+l !'

{ j k.1,IO<j+k+Z
Thus l:Singr(Si),Pi IIlilllt be one of (Xi, Yi, zt), (Xi, zd, (Yi, Zi) or (XiYi, Zi). Singr(Si) has SNCs at Pi unless y'] = (XiYi, Zi) is the completion of an ideal of an irreducible (singular) curve on Vi. Let E = 7r;1(pi_d rv p2. By construction, we have that Xi = 0 or Yi = 0 is a local equation of E at Pi-I' Without loss of generality, we may suppose that Xi = 0 is a local equation of E. Zi = 0 i:s a local equation of the strict transform of z = 0 in Vi Xv spec(R), so that (Xi, Zi) is the eornpletion at qi of the ideal sheaf of a subscheme of E. Thus there is a (,~urve C in Vi such that IC,Pi = (Xi, Zi) by Lemma. 5.3. The ideal sheaf :1 = (ISingr(Sd : Ie) in (JVi is such that (:1 n IC)Pi ~ ISingr(Sd,p" so t.hat (XiYi, Zi) is not the completion of an ideal of an irreducible curve on Vi, and Shlgr(S'i) has simple normal crossings at.

PiWe may thus assume that rno is 8ufficently large that Singr(Si) has SNCs everywhere on Vi for i > mo. We now make the obRervation that we must have bji # 0 and aid i= 0 for SOlne j and k if i > mo. If we did have bji = 0 for all j (with a similar

analYlSis if U'ji, = 0 for all j) in (5.4), then, since ISingr(S'i),Pi

= v'J,

5.1. Resolution of surface singularities

53

where J is the ideal in Ri, generated by lJ.i+k:+lj{,

j

k

ttIO<j+k+-l
lJx.8y.8z. t j t

we have (with the condition that bji = 0 for all j) that .jJ = (Xi, Zi). Since Sing,. (Si) JUtS SNCs in Vi, there exists a non-singular curve C C Singr (Hi) such that Xi = Zi = 0 are local equations of C in Ri,. Let)' : W ~ Vi be the blow-up of C. If q E ).-1 (Pi), we have regular parameters (u,v,y) in 6~~q such that Xi =U, Zi = u(v + (3) or Xi = UV, Zi = v. If ff = 0 is a local equation of the strict transform Sf of Si in W, we have that lIq (f') < 'I' except possibly jf ;r:i = 'I", Zi = 'lW. Then

I' = We see that

f'

it + a2fuQ,2i-2vr-2 + ... + o,7·(U ad -

r•

is of the form of (5.4), with bji = 0 for all.i, but

filin {a~i I 2 < j < 'f'} J has decrea..qed by 1. We see, using Lelluna 5.4, t.hal, after a llnit.e number of blow-ups of non-singular curves in Singr of the strict transform of S, the multiplicity must be < r at all points above Pi. Thus Pi is a good point, a contradiction to our a.."sumptioll. Thus b ji =1= () and o,k'i =1= 0 in (5.4) for some j and k if i > mo.

If fol)ows that, for i > mo, we must have •

or Yi = Yi+l, Thus in (5.4), we Illust have for 2 < j < r, Xi

(5.5)

= Xi+lYHl,

aj,i+l

= aji + bji -

aj,i+J = Q.ii,

j,

Zi

= Yi+l Zi+l·

bj,i+l = bji

bj,.f.+ I = aj,i

+ bji -

or

j

respectively, for all i ;?: 'Tno.

Suppose that ( E R. Then the fractional part of ( is

{(} = t if ( = a + t with a E Z and 0 < t < 1. Lemma 5.13. For i > mo, Pi is a good point on that aji=f 0 and

Vi

if there exists j such

5. Surfa.ce Singularities

54 1. aji

<

aki

bji

<

bki

j-k' j-k for all k with 2 < k < T and aki =I- 0, while 2.

Proof. Since Vp.j(Si) = r, aji +bji > j. Condition 2 thUR inlplies that either aji > .1 or bji > j. Wit.hout loss of generality,aji > j. Then condit.ion 1 implies that aki > k for all k, and ISingr(Si),Pi C

(Xi, zi)Ri.

Since Singr(Si) has SNCs, there exists a non-singular curve 0 C Singr(Si) such that Ie, Pi = (Xi, Zi). Let 1r : V' = B(G) ~ Vi be the blow-up of C, and let Sf be the strict transform of S on V'. By LeIIlIua 5.7, Singr (8') has SNCs. A direct local calculation ShOWfl that there is at most one point q E 1r l(pd flnch that lJq(S') = r, and if such a point. q exists, then OVI,q has regular paranleters (:1;1, y', z') such that "-

x..• = x,I A local equation

f'

y.- y I I

• t -.

Zi

= .r., I z I .

= 0 of S' at q is

.f' = (ZIt + U2i(x , )a2i -2(yl)b:l (ZI)f-2 + ... + ari(x')(tr ,-l' (yl)hr i . t

Thus we have an expression of the form (5.4), and condition.~ 1 and 2 of the statement of this leuuna hold for f'. After repeating this procedure a. finite unmber of times we will construct V -.. Vi such that I1a (S) < r for all points b .. a on the strict transfonn S of S such that a maps to Pi, SlnGe Inust drop by 1 every time we blow up. 0 -

•

7 +7 (L"

Lemma 5.14. Let

Then

and if Ojki < 0 then <5JOk ,l'+1 -

1

8J'k l > -. r4 °

Proof. We nlay assume that the first case of (5.5) holds. Then Dj,k;i+l

. (bit bki)2

= OJ,k,i + j - k

5.1.

Resolution of surface sinKUlarities

If 6'k' ). ,t

< 0 we.must have ~ . 1 - ~...J. k r O. (

b.. _bk' ..I!:... _1, k ) .1.

2

55

Thus

> 1 > _1

- .1'2 k 2 - r 4 '

o Corollary 5.15. There exists

ml such that i

>

ml implies condition 1 of

Lemma, 5. 13 holds.

Proof. There exists ml such that i > ml implies tSjki > 0 for all .i, k. Then the pairs (at" b~l) with 2 < k < 'r are totally ordered by <, so there exists . . I eemeth, 1 4 (~ ~) a nllfllma j' j .

Lemma 5.16. There exist.~ an i' Lemma 5.13,

D

>

17~1

such that we have condition 2 of

Proof. A calculation shows that

~f t {b"} J1 > 1 {a"} implies

D

With the above i', Pi' is a good point, since Pi' satisfies the conditions of Lemma 5.13. Thufol t.he conclusions of Theorem 5.12 must hold. Now we give a the proof of Theorem 5.2. Let T' be the nmxirnal multiplicity of points of S. By Theorem 5.12, there exists a finite sequence of blow-ups or point.s V'" --+ V RIiCh t.hat all points of Singr (8m ) flIe good points, where 8 m is the strict transform of Son Vm . By Lemma 5.10, there exists a sequence of blow-ups of non-singular curves Vn --+ 'V;n such that Singr(Sn) = 0, where b'n i~ tIle lStriet tra.nsforrn or 8 on v,t. By defolcfmrling induction on r we can reach the case where Sing2(Sn)= 0, so that Sn is non-f;illgu!ar.

Remark 5.17. Zariski discusses early approaches to resolution in [91]. Some early proofs of resolution of surface singularitie:; are by Albanese [12], Bcppo Levi [61], Jung [57] and Walker [82]. Zariski gave several proofs of resolution of algebraic surfa.ces over fields of characteristic zero, including the first algebraic proof [86], which we present later in Chapter 8.

56

5. Surface Singularities

Exercise 5.18. 1. Suppose that S is a surface which is a subvariety of a non-singular Lhree-dimenflional va.rieLy V over an a.lgebraically elosed l1eld K of characteristic 0, and pES is a closed point. a. Show that a Tschirnhausen transformation of a suitable local equat.ion of' 8 at. lJ gives a f'ormal hyper-surface of rna.xirnal contact for S at p. b. Show that if f = 0 is a local equation of 8 at p, and (x, y, z)

are regular paraIueters at p ~uch that II(f)

thcn~:t'1 for

f .

= l/(f(O, 0, z)),

t = 0 is a hypersurface of maxiInal contact (A.20)

0 in a neighborhood of the origin.

2. Follow the algoritlml to Teduee the ITlultiplieity of f wherea+b>r.

= zr +a;Qyb =

0,

5.2. Embedded resolution of singularities In this section we will prove the followinp; theorem, which is an extension of the nlam resolution theorem, Theorem 5.2 of the previolls section.

Theorem 5.19. Suppose that S isa surface which is a subvariety of a nonsingular three-dimensional variety V over an algebraically closed field !( of nhwrru:te'l"i,l;jtir. O. Then then~ exists a seq·1J.e1Wf:~ oj m.onoidal I.nm..sjorrn,'1 07)C1' the singular locus of S such that the strict transform of S is non-singular" and the total transfoTm1T*(S) is a SNC divisor. Definition 5.20. A resolution datum 1(, = (Eo, E 1 , S, V) is a 4-tuple where Eo, El' 8 arc reduced effective divisors on the non-singular three-dimensional variety V such that. B - Eo U E 1 is a SNC diviHor. R is resolved at p E 8 if 8 is non-singular at p and E U S has SNCs at p. For 'f' > 0, let Sing7.('R) = {p E S 11Jp(S) > rand 'R is not resolved at pl. Observe that SingrCR) = Singr (8) if r > 1, and Singr('R) is a proper Zariski dosed subset. of S for 'f' > 1. For PES, let

1J{p) We have 0

= the number of components of E 1 containing p.

< 'fJ(p) < 3. For 1', t E ti,

let

Singr,t (1(,) = {p E Singr ('R.) 117 (p) > t}.

57

5.2. Embedded retsoluUon of singularities

Sing.r,lR) is a Zariski closed subset of Sing1.(R) (and of 8) . .Bbr the rest of this section we will fix

r = v('R)

t

=

= II(S, E) = max{vp(S) I pE S. R 'TJ(1<.) - the

is not resolved at p},

number of components of E, containing a point. of Singr(R). maximUlTI

Definition 5.21. A llermi..,..,ible transform of R is a I1JOIJoidal transform 7r : V' ---4 V whose center is either a point of Singr,t(R) or a non-sing1.dar curve C C Sing'f,t('R) such that C makes SNCs with g. Let F be the exceptional divisor Of7r. Then 11"* (E) uP is a SNC divisor. Vv"e define the strict transform 'R,' of 1<. to be 1<.' = (Eo, E}, 8'), where SJ is the strict transform of .9 ~ and we also define

Eo = 'rr*(EO)red + P, E~

E[l

-

Rl'.rict transform of El if lJ(S/~ 11""'((Eo

= 0, E~ = 7r*(Eo + El)red + F

+ flJdred + F) = v(R),

if 1/(8', 7f"'((Eo

+ El),wl + F) < II(R).

Lemma 5.22. WUh thr. notation of the previous dejinition, v('R')

< vCR)

and v(R') = l'(R) implies TJ(R') < 1/(1<.). The proof of Lemma 5.22 is a generalization of Lemmas 5.3 and 5.4.

Definition 5.23. P E Sing,,.,,(R) is a. IJregood point if the following t.wo conditions hold: 1. In a neighborhood of p, Singr,t(R) is either 8, non-singular curve through p, or two non-singular curves through p intersec..:tiJlg tra.nsversally at p (satisfy-ing 2 of Definit.ion 11.6). 2. Singr,I,('R.) makes SNCs with E at p, That is, there exist. regular parameters {x, 11, z} in OV;p such that the ideal sheaf of each COIIlpon~nt of E and each curve in Singr,t CR.) containing p is generated by a subset of {~l:,'Y, z} at Ii.

pES is a .good point if p is a pregood point and for any sequence 11" :

X n ---; X n - 1

---; •.. --.

Xl

-.

spec( OV,p)

oj' permissible monodial transforms centered at curves in Singr,t (Rd, where n'i is the transform of 'R on Xi, then q is a pregood point for all q E 7['-1 (p). A point p E Singr,t(R) is called bad if p if:! not good. Lemma 5.24. The 'number of bad points in Singr,l('R) is jinUt:~.

5. Surfa.ce Singularities

58

Lemma 5.25. Suppose that all points of Singr,t('R.) are good. Then there ea.:i.'Jts a sequence uJ permissible munu'idal t1'ansjunns, 1[" : V' """'+ V, ce.ntered at non-singular curves contained in Singr ,t(Si), where Si is the strict transform of S on the i-til. monoidal transform" such. that if'R,' i.r; the strid transj'U1'Tn of n. on V', then either

vCR.') < ll('R.) (}'I'

v(1?,')

= 11(1?,) and'T/(1?,') < 'T/(1?,).

The proofs of the above two lemmas are shnilar to the proofs of T..Iemma.., 5.11 and 5.10 respectively, with the use of embedded resolution of plane curves. Theorelll 5.26. Suppose that 'R is a 1'esulutiun datum such that Eo =

0.

Let

(5.6)

• • •

~

II" 11'11 IT V 71 ~ v '1£-1

~

..•

7T"1

~

V;o-=- V

be the sequence where 1rn is the monoidal transform centered at the union of bad points in Singr,t CRi).. where ~ is the transform of n on Vi. Thwn I,his sequence 'is finite, so that it terminates in a Vm such that all points of

Singr,t('Rm) are good. Proof. Suppose that (5.6) is all infinite sequence. Then there exists an infinite sequence of points Pn E.. Vn such that 1l'n+l (Pn+d = Pn for all nand Pn is a bad point. Let R,t = C'JVn"Pn for n > O. Since the sequence (5.6) is infinite, its length is larger than 1, so we may assume that t < 2.

Let f = () be a local equation of S' at Po, and let Uk, 0 < k -s: l, be local equations of the components of E 1 containing Po. In Ro = OVtP there are regular parameters (x, y, z) such that v(f(O, 0, z)) = rand V(gk(O, 0, z)) = 1 for all A:. rrbis foHows sinee for each equation this is a 1I011-t.rivial Zariski open condition on linear changes of variables in a set of regular parameters. By the Weierstrass preparation theorem, after multiplying f and the gA: by units and performing a Tschirnhausen transformation 011 J, we have exprelSlSioll8

(5.7)

f gA:

=

=

z Z

'I'

~r

(

+LJj=2 aj x,y

)

r-j

z , ~ .. ¢k(X, y), (or gA; is a unit)

in Ro- z = 0 is a local equation of a formal hypersurface of maximal contact for = O. Thus we have regular parameters (XI, yI, Zl) in R 1 = C'JV'],Pl defined by

r

with a E [(, or

A

5.2. Embedded resolution of singularities

59

The strict transforms h uf f and Ykl of 9k have the form of (5.7) with aj replaced by :f (or ~), q>k wit.h ~ (or ~). After a finite number of Xl

L

Yl

l>'l

monoidal transforms centered at points Pj with v(!J) = r, we have forms in Ri, = OV;,Pt for the strict transforms uf f and Yk: ~

(5.8) such t.hat. aj, cPki are either units or zero (as in the proof of Theorem 5.12). We further have that loca.l equations of the exceptional divisor of Vi -+ V are one of Xi = 0, Yi = 0 or XiYi = O. This last case ca.n only occur if t < 2. Note that our a.qRumption "'(Pi) = t implies gki is not a unit in Rt for 1 < k < t.

Eb

We can further assume, by an extension of the argument in the proof or Theorem 5.12, tha.t aH irl'educible curves in Singr,t('Ri) are non-singular. By Lemma 5.14, we can assume that the set

l' =

{(a.~i b~i), I aj. ¥ J J 1

o}

U {(Cki,dki) I ¢ki

¥ O}

is totally ordered. Now by LeIIlruCl, 5.16 we can furt.her a..<;sume that

+ {b~Oi} < 1, { a~Oi} .10 Jo w Ilere

( aJUl~)' jo' jo IS t hc

n~=l 9i =

.. mInImum

0f

the (~~) j' j '

Reca11 that,i f t >, 0 0 makes SNCs with the exceptional divisor Eb of Vi -+ V. These

conditions imply that, after possibly interchanging the 9ki, interchanging ;r,'i, Y'i and lnllitiplying Xi, Yi by units in Ri' that the gki can be described by one of the following ca.ses at. P'i:

1. t = O.

2. t = 1, a. .q li = Zi + > 1. b. Y1i = Zi + xty/~, A > 1, J-L > l. C. 9li = Zi ~. t = 2, a. 9li = Z~, 92i = Zi + X·i·

xt, .\

b. gil = Zi + Xi, 9i2 = Zi + €Xi, where € is a unit ill R i such that the residue of f. in the residue field of Ri is not 1. c. gil = Zi + Xi, 9iZ = Zi + Exfyf, wbere f is a unit in R;,~ A > 1 and A + J-L ~ 2. If J.l > 0 we can take € = 1.

We ean check explicitly by blowing up permissible curves that Pi is a ~ood

point.

In this calculatiun, if t

=

0, this follows from the proof of

5. Surface Singularities

60

TheoreIIl 5.12, reealling that Xi = 0, Y'i = 0 or XiYi = 0 are local equations of Eo at Pi. If t = 1, we must remember that we can only blow up a curve if it lies on 91 = 0, and if t = 2, we can only blow up the curve gl = g2 = O. Although we are conlStructing a sequenee of pennissible monoidal transforms " over spec(Ov'tpJ, this is equivalent to constructing such a sequence over spec(OVi.,Pi)' as all cnrvp,s blown up in this calculation are actually algebraic, since all irreducible curves in Singr,tCR-i) are Ilon-lSingular. We have thus found a contradiction, by which we conclude that the sequence (5.6) has finite length. 0 The proof of Theorem 5.19 now follows easily from Theorem 5.26, Lemma 5.25 and descending induction on (r, t) E N X N, with the lexicographic order.

Rernark 5.27. This algorithm is outlined in [73] and [62]. Exercise 5.28. Resolve the following (or at. lea.qt make (v(R), 1](1(,» drop in the lexicographic order): 1. R = (0, E l , S), where S has local equation f = z = 0, and E 1 ilS the union of two components with respective local equations 91 -= z + a:y, [J2 = z + :r.i 2.

n

(0, El, S), where S has local equation f = z3

+ x 7 y7 =

0 and El is the ,mion of two components with respective local equations 91 = Z xV, 92 '= Z + y(:r. 2 + y3). =

+

----Chapter 6 Resolution of Singularities in Characteristic Zero

The first proof of resolution in arbitrary dimension (over a field of characteristic zero) was by Hironaka [52]. The proof CODl:lists of a local proof, and a complex web of inductions to produce a global proof. The local proof uses sophisticated methods in commutative algebra to reduce the probleul of reducing the multiplicity of the strict. transform of an ideal after a pernlissible sequence of monoidal transforms to the problem of redudng the multiplicity of the strict transform of a hypersurface. The use of the Tschirnhausen transformation to find a hypersurface of maximal contact lies at the heart of this proof. The order II and the invariant l' (of Chapter 7) are the principal invariants considered. De Jong's theorem [36], referred to at the end of Chapter 7, can be refined to give a new proof of resolution of singularities of varieties of characteri."tic zero ([9], [16], [70]). 'rhe resulting theorem is not as Mtrong as the classical results on resolution of singularities in characteristic zero (Section 6.8), as the resulting resolution X' ----4 X may not be an isomorphism over the non-singular locus of X. In recent years several different proofs of canonical resolution of singularities have been constructed (Bierstone and Milman [14], Bravo, Encina.'3 and Villamayor [17], Encinas and Villarnayor [40], [41], Encinas and Hauser [39], Moh [67], Villamayor [79], [80], and Wlodarczyk [84]). These papers

61

6. R.esolution of Singulal'iUes in Gha.l'acteristic Zero

62

have signHlcantly simplified Hironaka's original proofs ill [47] and [55], and represent a great advancement of the subject. In this chapter we prove the basic theorems of resolution of singularities ill cha.racteristic O. OUI' proof is ha.sed on nip.; a.Igorit.hms of [40] and [41]. The final statements of resolution are proved in Section 6.8, and in Exercise 6.42 at the end of that section.

Throughout this dut.pter, UJll~~ explidtely ~tated otherwise: we will assume that all varieties are over a field K of characteristic zero. Suppose that Xl and X2 are sllhschellle~ of a varip.ty W. WP. will denote Lhe reduced set-theoretic inter~ectionof Xl and X 2 by Xl nX:.!. However, we will denote the scheme-theoretic intersectibn of Xl and X2 by Xl . X2. Xl . X2 is the subscheme or W wit.h idea.l sheaf Ix. + IX2'

6.1. The operator 6. and other preliminaries Definition 6.1. Suppose that K i~ a field of characteristic zero, R = K[[xl, ... ,xnJJ is a ring of formal power series over K, and J = (fit ... fr) C R is an ideal. Define I

~

~

6(J) = 6 R (J) = (ft,···, fr)

) + (afi.. 8xj 11 < ~ < r, 1 <J < n .

"

6(J) is independent of the choice of generators eters :Vj in R.

Ii

of .1 and regular pa,ra.m-

Lemma 6.2. Suppose that W is a non-singular variety and .J c Ow is an ideal.t;heaf. Then there exists an ideal 6(J) = L\w(J) c Ow such that 6(J)Ow,q

= li(.!)

for all closed !winl"fj fJ E It'\l.

Proof. We define 6(J) as follows. We can cover W by affine open subsets

= spec(R) which satisfy the conclURions of Lemma A.ll, so dYIR e: ... €B dYnR. If r(u, J) = (il,· .. ,ir), we define U

r (U, 6 (J)) = (Ii, ; Ii I 1 < i < r, 1 < .i < n).

that

nh/K =

Yj This definition is independent of all choices made in this expression. Suppose that q E W i~ a closed point, and Xl, ... ,Xn are regular parameters in OW,q. Let U = spec(R) be an affine neighborhood of q as above. Let A = Ow,q. Then {dYl, ... , dYn} and {dXl, •. " dx n } are A-hases of n~/k by Lemma A.II and Theorem A.IO, Thus the determinant

1(:~)I(q) oF 0

6.1. TIle opera,tor 6. and other preliminaries

63

from which the conclusions of the lemma follow.

o

Given J c Ow and q E W, we define 1I.,(q) = lIq (J) (Definition A.17). If X is a subvariety of W, we denote v x = VI x' We can define ~" (J) for r E N inductively, by the formula ~r(J) = ~(~r-l(J»). We define t.1 0(J) - J.

Lemma 6.3. Supposc that W is a non-singular variet.y, (0) =1= J c Ow is an ideal sheaj, and q E W (which i.'1 not neces.9arily a closed point). Then: 1. l)q(J) = b > 0 ij and only if vq(6.(J))

2.

vq(.J)

3.

lIq (J)

4. v J

:

= b - 1. = b > 0 ij and only ij flq(~b-I(J» = l.

> b> 0 ij and only ij q E V(6. b- 1 (J)). W --+ N i.q o.n uppcr semi-continuous junction.

o

Proof. The lemma follows from Theorem A.lO.

If j' : X -+ I is a.n upper semi-continuous function, we define max.f to be the larp;est value assumed by j on X I amI we deflne a closed subset of X, Max j = {q E X I j'(q) - ma.x fl· We have Maxll,/ = V(~b-l(J)) if b _. max l)J. Suppose tha.t X shall denote by

i~

a closed subset of a n-dimensional variety W. We

R(l)(X)

(6.1)

c

X

the union of irreducible eornponents of X which have diIncnsion

11 -

l.

Lemma 6.4. Suppose that W is a Twn-sinylar' variety, .J c Ow i.e; an ideal .e;heaf, b = max lI.l and Y c I\rlax flJ is a non-singular subvariety. Let 7T" : WI --+ W be the mono'ido.l tran.9jorm with centcr Y. Let D be the

exceptional divisor of 7r. Then: O",~1

1. There is an ideal sheaf J 1 C -

such t.hat

b

J1 = I D J

and ID f J 1 · 2. If q E WI and p = 7r(q), then 11] 1 (q)

3.

maxflJ

<

1)J

(p) .

> max 1'71 .

Proof. Suppose that q E WI and p = 7T"( q). Let A be the Zariski closure of {q} in WI, and B the Zariski closure of {p} in W. Then A ~ D is proper. Dy upper semi-continuity of 1JJ, there exists a dosed point :z: E B such that vJ(x) = vJ(p), Let y E 1r- 1 (x) n A be a closed point. By semi-continuity

6. R.esolution of Singularities in Cll8.Tllctcristic Zero

64

of VJ1 ' we have VJ 1 (q) < V"J, (y), so it suffices to pl'Ove 2 when q and pare closed points. By Corollary A.5, there is a regular system of parameters Yl, ... ,Yn in CJw,p such that Yl = Y2 = ... = Yr = 0 are local equations of Y at p. By Remark A.18, we may assume that K = K(p) = K(q). Then we ean make a linear ehange of variables in Yl, . .. ,Yn to a..~~mme that there are regular parameters y 1, ... 1 Yn in OWl,q such that

Yl

fi,

= Yl,Y2 = YIY2,··· ,Yr = YIYr,Yr+l = Yr+l""

,Yn = Y71.·

By Rema.rk A.IS, if suffices to verify the

= 0 is a lncal equation of D.

conclusions of the theorem in the complete local rings

R -.:"'" O'W;p = K [[Yl, ... 1 Yn] and

S = OWl,q = K[[Y11' .. , Yn]]· Y c 1iaxvJ implies l.1"1(J) = b, where 1] is the ~eneral point ol'the component or Y whose closure contains p. Suppose that f E JR. Let t = I.lRU') ~ h, Then there is an expansion

f

_

-

L

il •• a~l, ... ,ln Yl

• ••

i

Yn-Til

in R, with {l.'i" ... ;i n E K and ail, ... ,in = 0 if i 1 + exist it, ... ,in with i 1 + ... + 'i n = t such that ait,

+ i1· < t. i n =1=

Further, there

O.

Tn S, we have Yt 1: f = -tf where

f 1-- ·

L

+---+i,.-t . . . -i, .. a . ~l ""~~II Yl

... Y''-~l.:n I'I.

is such that Yl does not divide ft. Thus we ha.ve vs(ft) < vR(f). In particular, we have Y~ I JS; and Rince there exists fER with VR(!) = b, y~'111 JS. Thms

J1

= 7-b, J..D •

satisfies 1 and

v-:J1 (q) < v.r(p)

.=

b.

o Suppose that W is a non-sing"u]ar variety and J c Ow is an ideal sheaf. Suppose that YeW is a non-singular l:mbvariety. Let Yl, ... , Ym be the distinct irreducible (eOIlI1ecterl) components of Y. Let 7r : Wl --) W be the monoidal transform with center Y and exceptiona.l divisor D. Let lJ = D 1 +.. .+Dm , where Di are the distinct irreducible (connected) components

6. .1. The operatOl' D. and other preliminaries

(j5

of D, indexed so that 1r(Di) = Yj. Then by an argument as in the proof of Lemma 6.4, there exist.s an ideal sheaf]l C OWl such that

(6.2) where J1 is such that for 1 < i < m we have In, t J 1 and r.·i = Ll11i (.1), with 71i the generic point of Yi. J 1 is called the weak transform of J. There is thus a function c on WI, defined by c(q) = 0 if q ¢ D and c(q) = Ci if q E Di, such t.ha.t (6.3) In the

~i luation

of Lemma 6.4, J 1 has properties 1 and 2 of that lerruua..

With the notation of Lemma 6.4, ~upp08e that. J = Ix is the ideal sheaf uf a subvariety X of W. Thc weak transform X of X i~ the lSubscheme of WI with ideal sheaf )1- \Ve see that the weak transform X of X is much easier to calculate than the strict tranlSfurm X or x. We have inclusions

x eX C 7l"*(X). Exanlple 6.5. Suppose that X is a hypersurface on a variety "'~ .,. max lJx and Y c lvlax l/X is a non-singular subvariety. Let 'rr : "(1 ---of> l-V be the monoidal transform of W with center Y and exceptional divisor E. In t.his (:&~e the strict transform X and the weak transforrn X are t.lua. same scheme, and 7r*(X) = X + rH;. Example 6.6. Let X C W = A 3 be the nudal plane curve with ideal Ix = (z, y2 - x3) C A: [:L: y, z]. Let m = (x, y, z), and l~t 1r : B = B(m) --+ A3 be the blow-up of the point m. The strict tranlSfunn X or X is nOll-singular. The most intereRting point in parameters (Xl, Yl ,Z"l) such that ,.,. --

d..

"'1 ,

..A,.

.X

is thc point q E

y = :l;t'Yl,

Then

IX,q = (Zl, y~ - xt}. In this examplc, X

I

X and

7r* (X)

are all distinct.

1r- 1 (m)

with reg'ular

66

(j.

Resolution of SinguJa.rities in Ch8,racte";stic Zero

6.2. Hypersurfaces of maximal contact and induction in resolution t.ha1.W is a non-singular variet.y, .Ie Ow is an ideal sheaf and bEN. Define Stlppo~e

Sing(.1,lJ) = {q E W

I vJ(lJ) > b}.

Sing(J, b) is ZariBki closed in W by Lemma 6.3. Suppose that Y C Sing(J, b) is a non-sinp;ular (but not necessarily connected) subvariety of W. Let 1rl : W, ·~W be the lllonorlial transform with cen~er Y,and let D1 = 7r- I (y) be the exceptional divisor. Recall (6.3) that the weak transform J 1 of J is defined by

JOWl

= :Ttl Jb

where c is locally constant on connected components of D I . We have c .> b by upper semi-continuity of I)J. We can now define J1 by

(u.4)

JOWl =

:TAl .Ill

so that Suppose that

(6.5) is a sequence of monoidal transforms such that each 7ri is centered at a non-singula.r subva.riety 1i C Sing(Ji , b), where J t iB defined inductively by Ji-IOWi =

Th ..Ii t.

and D.i is the exeepUonal divisor of ?ri • We will say that (6.5) is a resolution

of (W, .I, b) if Sing(Jn , b)

= 0.

= max 1).1,

q E Singe.!, r). A non-singular eodimension 1 subvarietyH of an affine neighborhood U of q in W il:.i called a hypersurface of maximal contact for J at q if

Definition 6.7. Suppose that r

1. Sing(J, r)

n U c H,

and

2. if

Wn.

~

...

~

WI

~

W

is a sequenee of rnonoidal transfurrlls of the form of (6.5) (with b = r), then the strict transform lIn of lIon Un = W n Xw U is such that Sing(.!n., r) nUn c H n .

6.2. Hypersurfaccs of maximal contact and induction in resolution

67

With the assumptions of Definition 6.7, a non-singular codimension 1 subvariety H of U = spec( OW,q) iA called a fonnal hypersurface of maxinlal contact for J at q if 1 and 2 of Definition 6.7 hold, with U = spec( (')w,q). A

A

If X c W is a codimension 1 subvariety, with r = max 11X, q E max llX 1 then our definition of hypersllrface of maximal contact for J = Ix coincides with the definition of a hyper~urfaee of maxirnal cont.act for X given in Definition A.20. In this case, the ideal sheaf I n in (6.5) is the ideal sheaf of the strict t.ransform X71. of X on Wn • Now suppose that X c W is a singular hyper~UIfaee, and K i~ algebraically closed. Let 'f' = max{vx} > 1 be the set of poiIlt~ of maxhnal umltiplicity on X. Let q E Sing(Ix, r) be a closed point of maximal multiplicity r. The basic strategy of reAolutioll is to construct a sequence fVn

~

...

~

Wl

~

W

of monoidal transforms centered at non-singular subvarieties of the lOCUM of points of multiplicity r on the strict transform of X so that we eventually reach a situation where all points of the strict. transform of X have multiplkity < r. We know that under such a sequence the multiplicity can never go up, and always remains < r (Lemma 6.4). However, getting the multiplicity to drop to less Lhal ',. everyw here iR much more difficult. Notice that the desired sequence Wn --4 WI will be a resolution of the form of (6.5), with (.Ti, b) = (IXi' r), where Xi is the strict transform of X on Wi' Let q E Sing (Ix ,r) be a closed point. After Weierstrass preparation and performing a TAchirnhau.qen transformation, there exist regular parameters (Xl, ... , X n , y) in R = OW,q such that there exists fER such that I = 0 is a local equation of X at q, and ". (6.6) L- ar ( Xl,· .. 1 Xn ) Yr-i . f = Yr + ~ i=2

Set H = V(y) c spec(R). Then H = spec(S), where S = Kr[Xl, . .. , xnll. We observe that H is a (formal) hypersurface of maximal contact for X at q. The verification ~ ~ follow~. We will first show that II contains a formal neip;hborhood of the locus of points or maximal mnltiplicity on X at q. Suppose that Z C Sing (Ix , r) is a subvariety containing q. Let J = I Z,q' ~

Write J = P1 n· .. nPr ) where the Pi are prime ideals of t.he Aame height: in R. Let Ri = Rl'i' By semi-continuity of l1X, we have f E P[ RPi for all

6. R.esolution

68

i. Since {:J~

of Singularities in Characteristic Zero

Hpi -. Rp" we have (r -l)!Y Y E nJRg = J and Znspee(R) c H. :

= ~~1{

E JRpi for all i. Thus

Now we will verify that if Y c Sin~(Ix, r) is a non-singular subvariety, 7f : WI. -. W is the monoidal transform with center Y, Xl is the strict transform of X and ql is a closed point of Xl such tha.t ?T(ql) = q which has maximal multiplicity r, then ql is on the strict transform HI of H (over the formal neighborhood of' q). SUIJpOSe that q1 is ~uch a point. That is, 1 t]1 E: ?T- (q) n Sillg(Ixl' 1'). We have shown above that y E I y . Without changing the fonn of (6.6), we ma.y then a."lSlUUe that there is s < n such that Xl = ... = X s = Y = are (formal) local equatiollii of Yat q, and there is a fOrIna! regular systmn of parameters xI(l), ... ,xn(l),y(l) in OW1,Ql such that

o

(6.7)

:1:1

:r,(l),

Xi

:L:l(l)~:i(l)

Y a:,t

xI(l)y(l), :l:i (1) 1'01' ,c; < i < n.

for 2

< 'i, <

oS,

Sil1?;(Ix~ r)~ we have qi E (Xl, ... , xs)i for all 't.. Theil we havf: a (formal) local equation il = 0 of Xl at ql, where

Since Y

c

il -

~ = Xl

yU)T + ~ a,~ y(l)r-i ~xl i=2 1

.

1

(6.8)

.

'r

-

Y(l)T + Lai(l)(Xl(l),.,., x n (1))y(ly-i i-2

a

and 'Yl (1) is a local equation of HI at Ql, while ~: 1 (1) = 0 is a. local equat.ion of the exceptional divisor of?T. With our assumption that .ft also has IIlultiplidtyr, h has an expression of the same form as (6.6L so by induction, If is a (formal) hypersurface of maximal c.:ontaet. We Ree that (at leA.~t. over a fornlalncighborhood of q) we have reduced the problem of resolution of X to some sort or resolution problem on the (formal) hypersurfacc II.

We will now describe this resolution problem. Set ·,.1

d

I = (a2' ... ,a;) C S = !(llxl,'" ,xn]]' Observe that the scheme of points of multiplicity r in the formal neighborhood of q on X coincides witll the scheme of points of multiplicity > r! of I~

Sing(f R, r) = Singe!, r!). However, we may have VR(I) > 7'1. 011

Let us now consider the effect of the IIlonodial transform 'IT of (6.7) I. Since H is a hypersurface of maximal contact, 1r certainly induces

6.2. Hypersurfaces of maximal contact and induction in rcsolutioll

fm

a morphism HI -+ H, where HI is the strict transform of II. By direct verification, we see that the transform 11 of 1 (this transforlll is defined by (6.4) satisfietl

where 81 the ideal

= K[[a:1(1),·· .,:cn(l)ll h 81 =

=

OHhql' and this ideal is thus, by (6.8), r!

(a:.! (1) 2 , ..• , ar (l)

r! r ),

which is obtained from the coefficients of 11, our local equation 01" Xl, in the exact same way that we obtained the ideal I from our local equation .f of X. We further observe that Zlb\ (lISt} > r! if and only if Zlq1 (Xd > r. Vle see then that to reduce the nlllltiplidty of the strir.t t.ransl"orrn of X over q, we are reduced to resolving a sequence of the form (6.5), but over a formal scheme. The above is exactly the procedure whieh is followed in the proof of resolution for curves ~iven in Section 3.4, and the proof of resolution for surfaces in Section 5.1. When X is a curve (dilll 117 = 2) the induced resolution problem on the formal curve II is essentially trivial, as the ideal I is in fact just the ideal generated by a monomial, and we only need blow I1p point.s to divide out powers of the terms in the rIlonOInial. When X is a surface (dim W = 3), we were able to nlake use of another induction, by realizing that resolution over a. general point of a. l:urve in the locus of points of Illaximal multiplicity reduces to the problem of resolution of curves (over a non-closed field). In this way, we were able to redur.e to a resolution problem over finitely many points in the locus of points of maximal multiplicity on X. The potentially worrisome problem of extending a resolution of an object over a fonnal germ of a hypersurface to a global proper morphism over W was no(, a great difficulty, since the blowing up of a point on the formal surface H always extends trivially, and the only other kind of blow-ups we had to consider were the blow-ups of non-singular r.urves contained in the locus of maxirnal multiplicity. This required a little attention, but we were able to fairly easily reduce to a situation were these formal curves were always algebraic. In this case we first blew up points to reduce to the situation where the ideal I was locally a monomial ideal. Then we blew up more points to make it a principal monomial ideal. At till.., point, we finished the resolution pl'OrR~"8 by blowing up non-singular curves in the locus of maximal multiplicity on the strict transform of the surface X, which amounts to dividing out powers of the terms in the principal monomial ideal.

In Section 5.2, we extended this aigoritlull to obtain embedded resolution of the surface X. This required keepinf?; track of the exceptional divisors

6. ResolutioIJ of Singularities in Characteristic Zero

70

which Occur under the resolution process, and requiring that the centers of all monoidal transforms make SNCs with the previous exceptional divisors. We introduced the 77 invariant, which counts the number of componentl:l of the exceptionallocu8 which have existed since the multiplicity last dropped. In this chapter we give a proof of rel:lolution in arbitrary dimension (over fields of characteristic zero) which incorporates all of these ideas into a general induction statement. The fact that we must consider some kind of covering by local hypersurfaces which are not related in any obvioilli way is incorporated in the notion of General Basic Object.

In the algorithm of this chapter Tl:lchirnhausen is replaced by a method or Giraud for finding algebraic hypersurfaces of maximal contact for an ideal J of order b, by finding a hypersurface in Llb-1 (J).

6.3. Pairs and basic objects Definition 6.8. A pair is (Wo, Eo), where Wo is a non-singular variety over a field ]( of characteristic zero, and E u = {D11'" , Dr} is an ordered collection of reduced non-singular divisors on W o such that D1 + + Dr is a reduced simple normal crossings divisor. I

I

•

Suppose that (Wo, Eo) is a pair, and Yo is a nOll-singular closed subvariety of Woo Yo is a permissible center for (Wo, Eo) if Yo has SNCs with Eo. IJet 7r : WI -10 W be the monoidal transform of W with center Yo and ex.eeptional divisor D. By abuse of notation we identify Di (1 < i < r) with its strict transform on WI (which could become the empty set if Di is a union or components of Yo). By a local analysis, using the reg1Jlar parameters of the I!foof of Lemma 6.4, we see that. D1 + ... + D 1• + D is a SNU divisor on

"'1·

Now set El = {D 1 ,·.· 1 DT} lJr +l = D}. We have thus defined a new

pair (WI, E1), called the transform of (Wo, Eo) by the map

(Wl, Ed

-?

'fr.

The diagram

(Wo, Eo)

will be called a transformation. Observe that some of the strict transforms D i may be the empty set.

Defi.l!ition 6.9. Suppose that (Wo, Bo) is a pair with Eo = {D 1 , ••• , Dr}, and Wo is a non-singular subV'
-

(Uro, Eo) is a pair, and

CWo, Eo) ~ (Wo, Eo) is called an immersion of pairs.

71

6.3. Pairs and basic objects

Definition 6.10. A basic object is (Wo, (.10, fJ), Eo), where (Wu, E u) is a pair, Jo is an ideal sheaf of Wo which is non-zero on all connected components of Wo, and 1, is a natural number. The singular locus of (Wo, (Jo, b), Eo) is Sing(Ju, b) = {q E Wo IZlq(JO) > b} = V(~b-l(.Jo» CWo.

Definition 6.11. A bask object (Wu, (Jo, b), Eo) is a simple basic object if

1/,,(.1) = b ir q E Sing(J, b).

objeet and }ij is a non-sin~ular closed subvariety of Woo Yo is a permissible center for CWo, (Jo, b), Eo) if Suppose that (lVo, (Jo, b), Eo) is

II.

ba.~ir.

Yo is permissible for (Wo, Eo) and YO c Sing(Jo, b). Suppose that YO is a permissible center for (Wo, (.To, b), Eo). Let 7r : WI -+ Wo he the monoidal transform of Wo wit.h cent.er YO and exceptional divisor D. Thus we have a t.ransformation of pairs (WI, EI) -+ (Wo, Eo). Let]l be the weak transform of J o on WI, so that there is a locally constant function CIon D (constant on each connected component of D) such that the total transform of .10 is JO O Wl

=I6 J I'

We necessarily have CI > b, so we can define -b T P'l Iel -bJ J 1 = Tn tlovWI = D 1·

We have thus defined a new basic object (WI, (JI' b), Ed, called t.he t.ransfOTIn of (Wo, (Jo, b), Eo) by the map 7r. rrhe diagra,lU

(WI, (JI, b), El)

-+

(Wo, (.10 , b), Eo)

will be called a. transformation. Observe that a transform of a ba..,ic object b a basic object, and (by Lemma 6.4) the transform of a simple baJ=iic object is a simple basic object. Suppose that

(£i.9)

is a sequence of transformations. Definition 6.12. (6.9) is a resolution of (Wo! (Jo, b), Eo) if Sing(Jk, b)

.=

0.

Definition 6.13. Suppose that (6.9) is a sequence of transformaLious. Set J o = Jo, and let Ji+l be the weak transform of J i for 0 < i < k - 1. For o < i < k define

by w-ordi(q)

=

II

q

(J.) b

~

6. Resolution of Singul8,rities in Characteristic Zero

72

and def1ne by ord i () q =

lJq(Jd b .

If we assulIle l.ha,l Yi CMax w-oro'i C Sing(.h, b) for 1 < i

< k in

(6.9),

then

nulX w-ordo > ... > max w-ordk by Lemma 6.4.

Definition 6.14. S\lppo~e that (6.9) is a sequence of transformations such that Yi c Max w-ordi C Sing( Ji , b) for 0 < i < k. Let ko be the smallest index such that maxw-ordko-l > max w-ordko = = maxw-ordk' In particular: ko is defined to be 0 if rIla.x w-orr1o = = max w-ord k . Set Ek = E~~ U Ek, where E}; is the ordered set of divisors D in Ek which are strict transforms of divisors of Eko a.nd Et - Ek - E;;. Define I

tA: :

Sing( Jk I b)

---+

CQ x Z,

where Q x Z has the lexicographic order by tk(q) = (w-ordk(q), 7J(q»:

_ { Card{D E Ek I qED}

7}(q) -

Card{D E

E;;

I qED}

if w-ordk(q) if W-ordk(q)

< nlaxw-ordkul = lIlax

w-nrd ko .

This definition allows us to inductively define upper semi-continuous functions to, tt, ... I tk 011 a sequence 01' 1,ransl'onna1.ions (6.9) snch that. l'i c Max w-ordi C Sing(Ji, b) for 1 < i < k.

Lemma 6.15. Suppose that (6.9) is a sequence ojtransjormations such that li C Maxti C l\1ax w-ord.; /01' 0 < 'i, < k. rlum, max tk

< ... :5

max tk·

Lemma 6.15 follows I'rom Lemma 6.4 and a local analysis, llsing the regular parameters of the proof of Lemma 6.1.

Definition 6.16. Suppose that (Wo, (Jo b), Eo) is a basic object, and l

(6.10)

(Wkl (Jkl b), E k ) ~ ...

---+

(Wo, (Jo, b), Eo)

is a, seqIlence of transformation..c:; such that max w-ord~: = O. Then we say that (Wk1 (Jk, b), E k ) is a monomial basic object (relative to (6.10). Suppose that (WA:, (.h, b), EAJ is a monomial basic object (relative to a sequence (6.10»). Let Eo = {Db"" Dr} and let D i he t.he except.ional divisor of 1ri : Wi ---+ Wi-l, where ~ = i + r. Then we have an expansion (in a neighborhood or Sing(Jk' b)) of the weak transform .fA: of ./:

Jk =

zdJ. .. .~~, 1

Ie

6.3. Pairs and basic ohjects

73

where the dr are functions on Wk which axe locally constant function on IJi for 1 < i < k.

~el'o

on

WA:

Lht and d-:f is a

Suppose that B = (Wk: (Jk: b), E k ) is a monomial basic object (relat.ive to (6.10)). We define reB) : Sing(Jk, b)

-7

1M = Z >< Q >< ZN

by where

rl (q) = min

p such that there exist il, ... ,'il " with 'r- < il, ... ,ip < r + k, such that diJ.(q) + ... + (~p(q) > b for some q E Dh di1 (q)+ .. +dip{q) . t - q-lJ ' WItll

'f'

n··· n V.i.." + k, _ r < 1.1, ••. ,1. p < •

.

such that p = rl (q), di i (q) + ... + dip(q) > b, q EVil

r3(q) -= max

"

"

with r


(lI, ... ,1.1'>:

0,

,

.

,

n·· -n Dip

_ dil (q) '---+diP.(q)

) such that r 2(q) ,';'p < 'r + k, q E DiJ.

b

'

n· .. rl Dip

Observe tha.t i'(B) is upper semi-continuous, and Maxf(B) is non-singular.

Lemma 6.17. Suppose that D -= (Ulk, (Jk : b), Ek) is a 1nonomial basic object (relative to a sequence (6.10)). Then}'k = Maxr(B) is D. permissible ce.nleTjor B, and if (Wk+l, (Jk+ b b), Ek+l) --) (~Vk, (Jk, b), E k) is the lrlLn.,,formation induced by the monoidal transfonn 7rkll : Wk:+ 1 . ~ Wk with center Yk, then Dl = (l--Vk+l, (Jk +1 , /)), E'f.:+') is a monom-ial basil: ulJjed and max r( B) > maxr(Dl)

in the lexicographic order. Proof. Max r(B) is certainly a permissible center. We will show that

maxf(B)

> Iuaxr(BJ).

We will firl:lt assume that nlax(-rtJ < -1. Let. 1. = i+ r. There exist locally constant functions di on W such that d:;:(q) = 0 if q ¢ ~. and dr is locally eonstant on Di, such that -dr .. . -dk Jk• -- -.L [), 1-D7' 1 k

Let. D I be the strict transform of D l on W k+ 1 and let D k+ 1 be the except.ional divisor of the blow-up 1rk+l ; Wk+l ~ Wk of Yk = Maxr(B). Then

-aT rrdk+l Jk+l - ~ ... ~... , D1

Dk+J.

74

6. Resolution of Singularities in Characteristic Zero

where for 1 < i < k we have dr(q) = £4(11"k+l(q)) if q E Dr and di(q) = 0 if q (j. Di' We have

dk+ 1(q)

= dh (11"k+ 1( q)) + ... + dip (11"k+ 1 (q)) -

b

°

if q E D k+ l' and dk +1 ( q) = if q '/. 15k +! . For q E Sing(Jk+ll b) we have r(BIJ(q) Maxr(B), so it suffices to prove that

= r(B)( 11"k+l (q))

if 1I"k+l (q) ¢

r(DI)(q) < r(D) (1I"k+l (q))

if qE Sing(Jk+b b) and 1I"k+I (q) r(Dl)(q)

E

Maxr(D). Suppose that

(-p1, Wl, (j1, . .. , .iPl' 0, ...))

=

and

r(B)(7T"k+l(q)) = maxr(B) = (-p, W, (il" .. ,ip , 0, ... »). We will first verify that

PI = rtCq) > rl(7T"A~+l(q») = p.

(6.11)

If k + 1 ¢ {jl, ... ,.iPl}' then the inequality b<

iInplies p,

djl

(q)

+ ... + djpl (q) <
> 1).

Suppose that k + 1 E {jl,' .. ,iPJ}' so that .il = k + 1. If PI exists ik E {iI, .•. ip} such that ik ¢ {jt, ... ,jpl}' Let

< P, there

I

d = dj] (q)

+ ... + djpl (q).

Then

d

dJ2(q) + ... + djpl (q) + di ] (7T"k+l(q» + ... + dip (11"k+l (q)) - b < (dil (7rk+I (q)) + ... +dik__ l(7T"k+l(q)) + dikll(7rk+l(q)) + ... + dip (7T"k+ l(q» - b) +diA, (7T"k+l (q)) + dh (11"k+l (q)) + + djp! (1I"k+l (q)) < dik(7rk+l(q») + d h (7rk+l(q)) + + dj'(lj (11"k+l(q») < b,

a contradiction. Thus IJI ?-- p.

=

p.

We will verify that WI

then

i1

+ 1. We must have + ... +dJT'J (11"k+l(Q)) < b,

Now assume that PI {jI , ... ,jPl }, we have

If A: + 1 E {il,'"

,iP]},

dh (7T"k+l(q»

= k

<

W.

If k

+1

¢

75

6.4. Basic objects and hypersurfaces uf maximal contact

so that w1b -

(dit (7I"k+I(q)) + + dip (7I"k+l (q)) - b) + dh(q) < d'h (7r k-U(q)) + + dip (7T'k+I (q)) + (djz( 7rk+I(q)) + ... + djp1 (71"11:+1 (q)) - b)

+ ... + dj

1'l

(q)

< 'wb. Thus Wl < 'lV. Assume that P = PI and {jIl ... ,jpl} in this case. Since D·H we must have (jI, ... ,jp,Ol"·)

'W

=

'WI.

We have shown that k + 1 ¢

n .. ·nD·p =0 ' < (il, ... ,ip ,O, ... ). 't

It. remains to verify the lemma in the cage when max( - r 1 (B)) This case is not diflicult, and is left to the reader.

= -1. 0

6.4. Basic objects and hypersurfaces of maximal contact Suppose that B = (W, (J, b), E) is a ba.'1ic object and {W'\} >'EA is a (finite) open cover of W. We then have associated bask objeds

B>'

= (W\ (.1\ b), E>')

for A E A obtained hy restriction to W>'. Observe that Sing(J\ b) = Sing(J, b) n W>..

= (WI, (JI, b)

E 1 ) --+ (W, (.1, b), E) is a transformation induced by a, monoidal transform 71" : WI -+ W, then there are associated transformations

If D1

l

(wt, (.1f,b), Ef)-) (w >. , (J >. , b) , E>'), where W1A = 7I"-I(W A) and Jt = J1 I W{ In particular, {W 1>'} Bt =

is an open

cover of WI. Definition 6.18. Let WI = W x A1- with projection 1T' = WI -+ lV. Sup-pose that (W; (J, b), E) is a basic object. Then there is a basic object (WI, (Jl, b), Bd, where .11 = JOWl' If E = {D 1 , ... , Dr} and D~ = 71"-1 (Dd for 1 < i ~ r, then E 1 = {D~ 1 . . . , D~}. The diagram (WI, (JI' b), E 1) -+ (W, (J, b), E)

is called a restriction. Remark 6.19. Suppose that (W, (.1, b), E) is a basic object and Wh C W is a non-singular hypersurfacc. Suppose that IWh C I:-:. b- 1 (J). Then Sing(J, b) c V(~b-l(J)) C Wh, and vq(l:-:.b-I(J) = 1 for any q E Sing(J, b).

Conversely, if (W, (.1, b), E) is a simple basic object and q E Sing(J, b), thenvq(~b-l(J)) = 1, so there exist an affine neighborhood U of q in W and a non-singular hypersurface Uh C U such that U n Sing(.I, b) c Uh.

6. Resulution of Singularities in Characteristic Zero

76

Lemma 6.20. Suppose that (WI, (Jl, b), Ed is a

WI

--+

(W, (J, b), E)

traw~forma.tion

~

of bl~sic objects. Let Dbe the exceptional divisor of W. Then for all integers i with 1 < i < b we have b .

.'

~w I(J)OWl C

and

1'1)

-l-~t,i(J)OWl C ~tv/(·h).

:PD

Proof. Let Y

C Singe J,

b) be the center of 1f :",'1

The Hrst, inclusion follows since Vq(~tvi(J)) divides 6 ~"i(J)OW1'

--+

W.

> i if q

E Y, so

that

Ib

Now we will prove the second inchlRion. The second inclusion is trivial if i = b, Rince JOw, = ItJl by definition. We will prove the second inclusion by descending induction on i. Suppose that the inclusion is true for some i > 1. Let qED be a dosed point, P = 7r(q). Let K' = K(q). By Lemma A.1 U, i L 8uffiees to prove the inclusion on W x K K', so we may assume that. K (q) = K. Then there exist regular parameters :r.o, .1:1, ..• ,:tn in R = OW,p such that Iy,'P = (XO, Xl, ..• ,:r. m ), with Tn < nj Xu, ~, ... , .~~ ,Xm+l, ... , Xn, is a re(?;ular system of parameters in R 1 = OWI,q, and ID,q = (xo). It suffices to show that for f E ~~ (i-1)(J)'Pl

!

(6·.1 2)

1-1

E Ah-(i-1)(J ) . DWI

IO

1 q

If f E ~tv'i(J)p c ~~ (i-l)(J)p, then the formula follows by indudion, ISO we may assume that f = D(g) with 9 E ~t,.i(J)p, D an R-derivation. Set D' = xoD. D' is an HI-derivation OIl Rl = OW1,q since D'(i:i) E OWl,q for 1 < ';' < Tn. By induction, 9

xi E

o

n]

A

b-i(J)""

Uw

D

Thus

A

b-i(J).

VW1,q C uWl

1 q

A

b-(i-l)(J )

C UWl

1 q'

D'($-) E ~ b-(i-l)(J ) , ~ WI 1 q, ·r.o

D'(L) 1

Xo

= D.(g) _ ~-l

iD(x

Xu·

U

).!L.

t'

Xo

which implies

f . _ D(g) _ D'(JL) + 'D( ) 9 x - -.. xi Z Xo xi

i-1 -

Xo

and (6.12) follows.

i 1

'0

0

E

~b'-(i-l)(J ) Wl 1 q,

0

D

77

6.4. Basic objects and hypcrsurfaces of maximal contact

Lemma 6.21. Suppose that (W, (J, b), E) is a simple basic object, and Wh C W is a 1wn·.tfli1l,g1.Llar hyper.91.Lrjace such that XWh. C ~ h-l (.1). Suppose that

(Wl,(,h,b),El)

-4

(W,(.J,b),E)

'is a tro,11M9jormation 'lllith center Y. Let (Whh be the strict transform of Wh on WI. Then Y C Wh, (Whh is a non-singula1' hype.'rS'lt1jace. in WI, a.nd IC Wh)1 C ~b-l(Jl). Proof. Let LJ be the exceptional divisor of qED is a closed point, and

1r :

Wi --. W. Suppose that

p = 7l"(q) EYe Sing(J, b) = V(L\h-l (J)) CWh. Recall that Y C Sing(J, b) by the definition of transformation. Let j = () be a local equation for Wh at p, and .9 = 0 a local equation of Dat q. Then ~ = 0 is a local equation of (WIt)l at q, and E .6. b- l (Jt}q by Lemma 6.20. Thus

f

D Definition 6.22. Suppose that B

= (W, (J, b), E)

is a simple basic object and W is a non-singular closed subvariety of W such that Xw c ~ h-I (J). Then the coefficient ideal of B on W is .

-

b-l ,",,'

hI

O(.J) = ~ .6.~(.l) b':, O~r. i=O

Lemma 6.23. With the notation oj Le.1nm.a 6.21 and DefinU:ion 6.22, Sing(J, lJ) -= Sing(O(J), b!) C Wit C W

Proof. It suffices to check this at closed points q E Who By Lemma A.16 we may assume that K(q) = K. Then there exist regular parameters Xl, ... ,X n at q in W such that Xl = 0 is a local equation for Wh at q. Let R = OW,q =

K[[Xl' ... ,xnl], S

=

OWh,q = K[[X2, ... , Xn]] with natural projection R --+ S.

It suffices to show tha.t (6.13)

VR(.l) > l} ifand only jf Lls(l.j(JS)) > h-j

for j = 0,1, ... , b- 1. Suppose that

f

E

j c R. Write

f = ~Uiu:~ i>O

with ai E S for all i. We have VR(!) ;? b if a.nd only if Lls(aj) j = 0, 1, ... , b - 1.

> h-

j for

6. R.esolution of Singula.rities ill Characteristic Zero

78

tFf • Thus We also have a,; E b.i(J)S, sinceaJ, is the projection in 8 of 4r J JI Xl J

a

.

we have the if direction of (6.13). The ideal ~) (j) is spanned by eleIIleIlt~ of the form

ai1 '·,·+i

. - () il "1.1 ,.··,~n Xl

with

f

E J and i l

f

--:---~~

(',

A

n

() i

.... Xnn

+ ... + in < j, and the projection of Cil,... ,i

.,u '1··

~i:l+·+in

n

in S is

.

all

..

f)X'Z,l ... aX~

Thu~

o

we have the only if direction of (6.13).

Theorem 6.24. Suppose that (W, (J, b), E) is a simple basic ob.ieet, and {W.\} .\EA is an open cover of W such that for each A there is a non-singular hypcrsu.rfaccW}~ c W.\ lw.ch thai, Iwt C 6,b-l(J>') and has simple

wt

normal crossings with E>'. Let E~ be the set of non-singular reduced divisors on W,~ obtained by inte'I'.filecl:ion oj E wil,'" TtV~. Recall that R(l)(Z) denotes the codimension 1 part of a closed reduced subscheme Z, as defined in (6.1). 1. If R = R(l)(Sing(J,b)) i 0, then R is non-singular, open and closed in Sing(J, bL and has simple normal crossings with E. Furthermore, the monoidal transform of W with center R induces a tran.9forrnation

(WI, (J1, b), E 1 ) such that R(l)(Sin~(Jl,b))

-+

(fir, (J, b), E)

= 0.

2. If R(1)(Sing(J, b)) = 0, for each A E A, the basic object

(wt, (C(J>'),b!),E~) ha.9 the following 1}rol}f~1'lie.'1: a. Sing(J>., b) = Sing(C(J>'), b!). b. Suppose that

(6.14)

(W.'l' (.lsl b), E s )

-+ ... -+

(W, (J,lI), E)

'is a sequence of transformations and restrictions, with induced sequences for eaeh A

(W: , (.J: , b), E:) Then

-+ . .. -+

1m' each. A there is

tions and restrictions

(W >., (J\ b), E>').

an induced sequence of transforma-

6.4. Basic objects and llypersurfaces of ma.xiInaJ contact,

79

and a diagTam 'lJJhe'l'f~ the veTtical a'T"l'0'lIJ.9 aTe clo.ged immeT.9ioru;

of pairs

(6.15)

(W;"E;) i

((Wh)~, (Eh)~)

-+

...

~

(W'\EA)

i

-+

-T

(W,t, E~)

.9uch thal Sing( C( JA )i, b!) = Sing ( J; , b)

for all i. Proof. \\'e may assume that W A = W. First suppose that T = R(l)(Sing(J, b)) i= 0. Then T C Sing(J, b) c Wh implies T is a union of connected components of Wh, so T is non-singular, open and dOl:ied in Sing(J) b),aud has SNCs with E. Suppose that q E T. Let f = 0 be a local equation of T at q in W. Since (~V, (J, h), E) is a simple basic ohject., J q = jbOW,q. [f'lrj : WI -+ W is the monoidaL transform with center T, then (Jl)qI = OW,qI for all ql E IVI such that 1I"l(ql) E T, and necessarily, R(l) (Sing(Jt, b)) = 0. Now suppose that R(l)(Sing(J, b)) = 0, and (6.14) is a, sequence of transformations and restrictions of simple basic objects. Suppose that (WI, (Jl , b), El) -) (W, (J, b), E) is a transformation with center Yo. By Lemma 6.21 we have }Q C Wh, the strict transform (lVh)l of Wh is a non-singular hypersurface in WI, and L( Wh) 1

c 6 b-l ( J 1) .

Since Wh has SNCs with E, Y C Wh and Y has SNCs with E, it follows tha.t Y ha...~ SNCs wit.h E U Who Thus (Wit) I ha...., SNCs with Ell and if (Ehh = E 1 . (Whh is the scheme-theoretic intersection, we have a closed immersion of pairs Since the ca....,e of a restriction is trivial, and each (Wi, (.h, b), E i ) is a simple basic object, we thus can conclude by induction that we have a diagram of closed immersions of pairs (6.15) such that

(6.16)

I(Wh);

c D,b-l(Jj)

for all j. We may assume that the sequence (6.14) consists only of transformations. Let ~ he the exceptional divisor of Wk -) Wk-l' It remains to show

that (6.17)

Sing(C(.J).j, b!)

= Sing(.rj, b)

80

(). Resolution of Singularities in Characteristic Zero

for all j. By LeUlIllR. A.16 and Remark A.IS we Illay assume that K is algebraically closed. For k > 0 and 0 < 'f < iJ- 1, set

[6

h i -

(JJL = 7~ [6 h- i (.T)L_ 1 0w" k

so that b

O(J)k

bl

L [L\b-i(.1)]:

=

O(Wh)k'

i=l

We will establish the following formulas (6.IH) and (6.19) for 0 < k .$ s and 0 < i < b - 1:

(G.1H)

Suppose that q E Sing(C( JO)k, b!) is a closed point.. Set Rk = OWk,ql 81.; = O(Wh)k,q' Then there arc regular parameters (Xk,l"'" Xk,n, Zk) in Rk sHch that L(WIl)k,q = (Zk), and there arc generators {iAJ of JkRk such that A

(6.19) with ak.u~ E K[[Xkl' ... ,:l:k,Tt]]

= Sk

such t.hat for 0

< a < b- 1

b!

(Uk.oJ b-c. E C(J)kSk' Assume that. the formulas (6.18) and (6.19) hold. ((i.18) implie~ 0(.1)11: C C(Jk ), so by Lemma 6.2:{, Sin~(Jkl b) C Sing(C(J)k: bI). Now (6.19) ~hows that Sing(C(.J)kl b!) C Sing(JA:' b), so that Sing(C(J)k, b!) = Sing(Jk, b), as desired. We will now verify Formulas (6.18) aud (6.19). (6.18) is trivial if k and (6.19) with k = 0 followM from the proof of Lemma 6.23.

= 0,

We now assume that (6.18) and (6.19) are true for k = t, and prove them 1'01" k = t + 1. Since

[.6. b- i (J)L C 6 b- i (Jd, we have

[6

b -

i(JJ]

1

t+

=

(7 1

Dt:rr

Ji

[6

b i -

(JJL C

(7

~Ji6b-i(JtJ c 6b-i(.11+1),

D f +1

where the last inclusion follows from t.he second formula of Lemma 6.20. \rVe have thus verified formula (6.18) for k = i. + 1. Let qt+l E Sing(C(Jo)t+l, b!) be a closed point, and qt the image of qt+l in {8ing(C(Jo)t,b!). in Rt which satisfy (6.19). Recall that we have reduced t.o the assumption that

By assumption, there exist regular parameters

(Xtll""

Xl n1 Zt)

6.5. General

ba~ic

objects

81

K is a.lgehraically closed. Let yt be the center of Wt+ 1 ~ W t . By (6.16), there exist regular parameters (:r.l' ... , x n ) in St = K[[Xtl,' .. I Xtn]] C R t such that (XI, ... , Xn , Zt) are regular para.meters in R t , there exists r with r < n such that Xl = ... = x 7• "='=' Zt = 0 are local equations of yt in OWt,qt, and Rt+l = OWt+l,<Jt+l has regular parameters (:J:t+ 1,1, ..• ,Xt+l,n, Zt+l) such tha.t. A

A

Xl :l:i

Zt Xi

-

Xt+l,l, :r.,+ 1, 1;1:t+ 1,i

for 2 < i < r,

Xt+l,lZt+l, Xt+l,i for r

< i < n.

In particular, Xt+l,l =: 0 is R local equation of the exceptional divisor Dt+l and Zt+l = 0 is a local equation of (Wh)tU' We have generators ft+l =

It

.b

:J: t.+ 1,1

~

= L..J atl l,uzf+ 1

if a < b, and

o 6.5. General basic objects Definition 6.25. Suppose that (Wo, Eo) is a pair. We define a general basic object (GBO) (,ro, Wo, Eo) on CWo, Eo) to be a collection of pairs (Wi, E i ) with closed sets Pi C vlli which have been com~t.ructed inductively to satisfy the following three properties: 1.

Fo c vllo.

2. Suppose that F i c Wi ha~ been defined for the pair (Wi, Ed. If '7ri+l : (Wi+b Ei+l) ~ (l-l~, E i ) is a rest.riction, then Fi+l 'lri1.\ (Fi)'

3. Suppose that Fi c Wi has been defined for the pair (Wi, BJ. If 7ri+l : (Wi +b Eill) ~ (Wi, Ei) iq a transformation centered at }Ii C Fi' then Fi+l C Wi+l is a closed set snch that Fi . Yi Fill - 1i'i+1 (Yi ) by the map 1i"i+l. r-.J

The closed sets Fi will be called the closed sets associated to (Fa, Wo, Eo).

6. Ret>olution of Singularities in Characteristic Zero

82

If

(WI, Ed

(Wo, E u) Fo, then we ca.n use the pains (Wi, E i )

-t

is a transformation with center Yo C and closed subsets }Ii C Wi in Definition 6.25 which map to (WI, El) by a sequence of restrictions and transformations satisfying 2 and 3 to define a general basic object (F1,WI,El) over (W1,El)' We will say t.hat. Yo ilS a pcrmissible eentm' for (Fo, Wo, Eo) and call

(Fl, WI, Ed

-+

(Fo, We, Eo)

a transformation of general basic objects.

If

(W"

Ed ~ (Wo, Eo)

is a restriction, then we can use the pairs (Wi, Ei) and closed subsets Fi defined in Definition 6.25 above which map to (WI, Ed by a sequence of restrictions and transformations satisfying (2) and (3) to define a general basic object (F1, WI, El) over (WI, El). We will call (Fl, WI, £',) ~ (Fa, Wo, Eo) a rcstriction of general basic: objeetlS. A composition of restrictions and transformations of general basic objects (6.20) will be called a. I1f;17ni.rssible sequence for the GBO (Fo, Wo, Eo). (6.20) will be called a resolution if Fr = 0.

Definition 6.26. Suppose that (Wo, Eo) is a pair, and (Fo,Wo, Eo) is a GDO over (Wo,Eo), with associated closed sets {Pi}' Further, suppose that {We} AEA is a. finite open covering of W o and dEN. Suppose that we have a collection of basic objects {B6} AEA with -" , (ao" ~ bA)~ Eo)· -·A Bo" = (W o

We say that {B6}AEA is a d-dimensional structure on the GBO (Fo, Wo, Eo) if for each ;\ E A 1. we have an immersion of pairs 'A (tiTA Jo: ...... 0 I E-0A)

where

2. if

---7

We has dimension d, and

(lATA E:l A ) YV 0' :.'0 ,

6.6. Functions on a general basic object

83

is a permissible sequence for the GBO (.ro, Wo, Eo) (with associated closed sets Jti C Wi for 0 < 'i < '1'), then the sequence of morphisms of pairs (W;,E;)-t ". ~ (Wt,E~) induces diagrams of immersions of pairs

(W;\E;) ~ ...

t

->. ->.

t

~

(Wr ,Er )

(wt,E6)

---+

. -.>. -.>.

(Wo , Eo )

---+

such that. ~. .>. .>. >. - > . - >. >. >. ~ >. , ) (W (a b ), E (W (0,0' b ), Eo ) , , r o r r

is a sequence of transformations and restrictions of basic objects,

and F( = Fi n for 0


w: =

Sing(a~, bA )

1',

6.6. Functions on a general basic object Proposition 6.27. Let (.1'0, Wo, Eo) be a GBO which admits a d-dirne.nsional structure, and fix notation a,9 in Definil.ion 6.26. Let Ft = Sing(aG, bA ).

The1'e is a function ord d : F u -4 Q such that for all A E A, the composition

l:">')0'>.

.['0

is the function ord>' : F~

-4

.

..cIa

~

-4

Q defined by Definition 6.1,9 for

= Sing(aG, bA ).

Ft >..

-4

.. d

ord tn\

L'

>.

>.

~>.

>.

->.

.

.

Proof. Let {BOJ.l.EA, where E u = (Wo ,(au, b ), Eo), be the gIven d-dImensional structure on (.ro, Wo, Eo). Let Eo = {DIl"" D7·}. We have closed immersions j6 ~ ~ for all A E A. Suppose that qo E Fo is a closed point, and A, fJ E A are such that there

wt wt

qS E wt and qg E wg such that ,i6(qS) = jg(qg) = qo.

are that

l/'l"

o

(u.S)

We must show

qg (ag)

1/

_ _-=-o---=-_

b>' bfI We will prove this by expressing these numbers intrinsically, in terms of the GBO (.ro, Wo, Eo), 80 that they are independent of~. We will carry out the ealculation for

B6.

Let (.1i, WI, El) ~ (.1"0, W o, Eo) be the re.~trictibn where Wi = W o xAk' Let ql = (qU' D) E WI, Let Ll = qo x Ak; then L 1 C FI = 1f)l(Fo),

84

6. Resolution of Singularities in Characteristic Zero For any integer N > 0 we can construct a permissible sequence of pairs

(6.21) where for i > 1, WH1 ...... Wi is the blow-up of a point. qi.' q'i is defined as follows. Let Li be the strict transform of LIon Wi, and let Dr be the exceptional divisor of n'i, where 1, = i + 7'. Then qi = Li n Di.

¥le have ql E Ll C Fl. We must have L2 C F'}. by 3 of Definition 6.25. Thus ?T2 induces a transformation of GBOs (F2, W2, E2) ----+ (F!! Wi, Ed, and q'l E F'l. By induction, we have for any N > 0 a. pennissibJe sequence (6.22) (6.22) induees a permissible sequence of b&~ic objects with and centers for i > 1:

q;

->.

>.

Observe that Li C

W/

(6.23)

>.

->.

(WN' (aN, b ), EN)

b' = IIqG (a6).

~

qf = (q6, 0)

7r ->. ,\ >. ->. ... ~ (Wo ,(uo l b ), Eo)·

for all i > 1, so that (].t E Wi>' for all i

>

1. Set

wt.

Suppose that (Xl, ... , Xd) are regular parameters at qB in Then (:1:1, ... , Xd: l) are regular parameters at qf in (where = spec(K[t])) , • - >. and thus there are regular parameters (xl(N),X2(N)"",Xd(N),t) ill W N at q~ which are defined by Xi = xi(N)t N - l for 1 < i < d. t. = 0 is a local equation of the exceptional divisor i>~ of TV~ ...... W~ -1 .

art

Suppose that I E O~x qX and vqx. (f) o 0 0 • expressIOIl I

f

L

=

ail, ... ,idXl1 ..

iI +.. ·id>r

where ail,... ,id an expanSiOn

E

K'. Thus in

'X~d

= r. Let K' =K(q6). We have an

E [('[lXII'"

j('[[Xl (N),

Ak

,Xd]] = 6Wo',q6~

... ,xd(N), tJ]

= 0w,\N' q,\N

we have

f = t(N-l),·

L

ail,... ,idXl(N)il

... xd(N)id + tN-ION

·il+..+id=r

Thus we have an expression Since 1~q~ ( as)

f

= t(N-l)rfN in 0W~A q ,where t N' N

= b', we have ao>'0WA

- I(N-l)b' T/>' .n. N:

b,\ N'N

fiN.

6.6. Functions on a general basic object

85

where IiJ~ f K~. We thus have an expression N

A _ Z(N l)(b' _.lr") T/A jj>' .n. N,

(6.24)

aN -

1il

and llq~(a~) > bA since q~ E F..rv

n W~

=

Sing(aRr,b A ).

Under the dosed irmnersioll W~ C W~ we have (forN > 1) iJ~ L71' tV~, where D~ has dimension d and is irreducible. Since FF, c tV~, it follows that dim(FN'

01)

= dim(F~' D N ) ~ dimb~ =

d,

and the following three conditions are equivalent: 1. dirn(FN' D N ) = dim(FF,' DjV) = d. -A A 2.D N cFN ·

3. (N -l)(b' - bA ) > bA• Thus ~ = 1 holds if and only if b' - bA = 0, which holds if and only if, for anyN, dim(FN . D N ) < d.. Thus the condit.ion 11q>' (

o

aa) -

bA

0'

-bA --1

is independent of A.

Now assume that b' - bA > O. Then, for N sufficently large,

(N - 1)( b - bA ) ~ b\ and FN

.

Drv =

iJ~ is a permissible center of dimension d for (FN, Wn , EN) .

We now define an extension of (6.22) by a sequence of transformations (6.25)

(FN+8, WN+S, EN+s) lI'~S

..•

1I'~1 (FN, WN, EN) ~ .. , ~ (Fo, Wo, Eo),

where '1rN+i is the transformation with center YN+i = FN+i . D N+i , and we assume that dim YN+i = d I'm' 'i = 0,1, ... , S - 1. Observe that. (6.26)

diIll YNH

= cl if and

only if' iJ~+i

-

lJ-.'V+? . W~+i C FN·/-'i'

We thus have YN+i = b~+i" (6.26) induces an extension of (G.23) hya sequence of transfonnations (6 . 27)

(A bA ) E) 'II"~.q -A (W N+S' aN+S' ,N+S ... ""~l - A ~ A lI'N ~ - A A d (WN' (aN' b ), EN) -+ (Wo , (ao 1 u .) 1 E u).

The centers of 7ri in (6.26) are iJ~ for j = N, _ J WN+i is the identity for i = 0, ... , S-1.

, N +S -1. Thus

IVN+i+l -+

86

6. Resolution of Singularities in Characteristic Zero

We see then that A _ I(N-l)W-b>')-ib>'}~A aN+i D->'_ "'-N N+i

for i = 1, ... , S. The statement that the extension (6.27) is permissible is precisely the st.atmcnt that (N - l)(b' - bA ) > Sb A , which is equivalent to the statement that dimFN+i . D N +i = d for i 0,1, ... , S - 1. This last statement is equivalent to

=

s<e = l(N-l)(b'-bA)J -

bA

N

t

where Lx J denotes the greatest integer in x. Thus iN is determined hoth by N and by the GBO (.ro, Wo, Eo). The limit

iN

b'

J~~N-l =-r;x-l is defined in terms of the GBO, hence is independent of A.

0

Theorem 6.28. Let (.Fa, Wo, Eo) be a GBO which adrnits ad-dimensional structure, and fix notation a.'J in Definition 6.26. Let Fi be the closed sets be the associated to the GBO. Let ~A = Sing(at,bA), and let w..:ord;, fund'ions of Definition 6.18 defined for F/'. Consider any permissible seq'lJ.ence of transformations

tt

(.1:,., W,.,Er ) ~ ... ~ (.ro, Wo, Eo)· 1. Then fm' 1 < i < r t there. m'e functions w-ordt : fi the composition .),

-4

to such that

.Jd

A Jt I:' w-nrn.. n Fi --+ L'i -+ "l:

Ft-t Q. 2. Suppose that the centers Yi of'lri+l are such that Yi c Max w-ordf. Then there aloe functions tt :Fi ~ Q x Z such that the composition is the function w-ordf :

is the function

Proof. Let Eo

tt :F'l

-+

Q x Z.

= {Dl, ... , Ds }.

Tbe functions ti are completely determined byw-ordf, with the cOll8traint that Yi c Max w-ordt for all i, ~o we need only show that w-ordf can be defined for GBOs with d-dimensional structure. The functions w-ord~ are equal to or~, which extend to ordo on Fa by Proposition 6.27. We thus have defined w-ordo = ordo. Let Dt be the exceptional divisor of 1T'i, where ~ = i + s, and let = Dr.

Dr

W/·

6.6. Functions on a general basic object

87

For ea.ch A and i we have expressions ..,...cl'" a·tA -_ . L : D'"

(6.28)

.

.-rCi>. -A

n>.i a·t '

".L~

1

at

where is the weak transform of 0.8 on functions on iJ~ We further have that

WiA, and

the

Cj

are loeally constant

J

Suppose that 'fJi-] is a generic point of an irreducible cornpoueut Ii' of Yi such thatTJi-l is contained in (and thus necessarily in W/), and that qi E Dt. is a point wWch we do not. require to be dosed, but is contained in = Dr, and 'lri(qi) ifol c.ontained in the Zariski closure of T/i }.

W/

Df wl·

From (6.28), we deduce that

(6.29)

Thus

L i

A

w-ord·t - 1 ('1]'t 1) '-.1.

(6.30)

1

. 1

A Cj

('1]i-l)

b>'

------

v. (7D~ - \) 71t -- 1

;)=

-1.

1

We further have

Suppose that q~ E Pi, and define qj for 0 < j < i···· 1 by qj To siInplify notation, we can assume that qj E Yj for all .i.

= 1r.i+l (qj+1)'

Let Vj be the j ('reducible component of Yj which contains be the generic point of Y.f for () < j < i - 1.

qj,

and let'1]j

Suppose that ..\ E A is such that qi E Wi>.. We have associated points qj and TJj in for 0 < j < i.

W/

We thus have sequences of points {qO, ql, ... , qi} and {'1]0, '1]1, ... , '1]i-l} with q;, TJj E

wf for 0 < j


and {

J J ' J '} -' h / , qO,Ql,· .. ,qi' } an d {·'1o,711,···,1/i-l WIt qj,TljE

W>. .< . j £or 0 < -J-~.

6. Resolution of SingulariUes ill Cll8.Tactcristic Zero

88

From equation (6.30) and indueLioTl we Ree that the values

~l(qj) I)>'

(6.31)

£or· J' - 1

-

,; I""

(,

and w-ord;(qi) are determined by: 1. The rational IJurnhers

(6.33) 2. The functions (6.31)

YH(q:,j)

={

if q~ E

~

if q~ fi

and Y:s c D y locally at qs, Ys or ~ ¢. D j locally at qf;'

Y:s

Our theorem now follows from Proposition 6.27, since w-OI'd;(fJi) is completely determined by the data (6,32), (6.33) a.nd (6.34). 0

Proposition 6.29. Let Do = (Fa, Wo, Eo) be a GBO wh'leh admit.q a. ddimensional stnu:l:nre. Fix notation as in Definition 6.26, Let

F/' = Conside.r a

pe.7·rni,'i,'1ilJlt~

HI' 8'lLCh

that max w-ordr

,Qeq1Lence of transformations

= (:F,., W Er ) 1"

11'~

= O. Then there

rr = r .'3·I.u~h

Sing(af,b A).

l'

•• ,

~

(Fa, Wo Eo) I

are functions

(8.,.) : F~. ~ JM = Z x

to X ZN

that the composition

is the function

of Defin'ition 6.16.

Proof. Suppose 1, hat. Eo = {Dl, .. , IDs}, The function objects (W;\ (a~, b>'), E¢) depends only on the values of

r

on the basic for 1 < i < r (with the notation of the proof of Theorem 6.28). Thus the conclusions D follow from (6.31) of the proof of Theorem 6.28.

p:

89

6.7. Resolution theorems for a general IJasic object

6.7. Resolution theorems for a general basic object Theorem 6.30. Let (:Fol~ Wo, E8') be a GBO with aS80ciatcd closed sets {Fl} 'wh:ir.h admits ad-dimensional .'itT'lLcture. Fix notation as in Definition 6.26. Let F~ = Sing(a)., bA ) = F!l n lV.A • t t 1 ·a

Cons'ide'l' a. permissible sequence oj tm,l1"qjormations (.r~, Wr , E:) ~ ... ~ (:F8,Wo, Eg)

(6.35)

such that the center Yi of 1T"i+l satisfies

Yi = O~ 1 ... , r

C Max (Hi. C ~1ax W-Ordi C

Fi

> O. Thi.", (~ondition implies w-ordi (q) < W-ordi-1 (1T"i ( q)), ti ( q) < for q E Fi and 1 < i < r. Thus for i

- 1 lmd maxw-ord r

maxw-ordo > maxw-ord 1 maxto > maxt l

> ... >

t'i-l ( 7r i ( q))

maxw-ordrl

> '" > maxtr .

Let R(l)(Max t1. } be the set of points whe,,.e Maxtr has dimension d - 1. Then R(l) is open and do.~p.d in l\faxtr and non-singular in W r .

0, then R(1)

i,'i a permissible center for' (:F;, W rl E~) with associated transformation

1. If H.(l)(Maxtr ) ;f-

1I'r+l "r'd ( .r r+ 1: ·W 1"+ 1. Ed) r+1 -+ r

and P.ithm' R(l)(l\tIaxtr+l)

(.-rod .I,;: , W 1', CJLld) r ,

= 0 0'" rnaxtr > maxtr+l'

2. If R(l)(Maxtr ) = 0, the1'e i.tiI a OBO (:F!-1, Wr,E:-l) with (d--l)dimensional .strlJ,(~t'lLre and associated closed set,,, {Fl- 1 } such that

if (fi.:l6)

(F;~r-l , WN 1 E~-l )-, ... ~

(n-

1

~ W n E:- 1)

is a. Te.solution (FI~ 1 = 0), thl:'-11, (6.tifj) induces a permissible. ,~e­ quence of t1Y:J,71"Q!ormations

(:J{ WN , E¥V) I

-+ ' . . -+

(.r:, w", E~)

such that the center Yi of 7ri+ 1 .'mtisfies

Yi c

J\,fax t'i C Max w-ordi C

Fid

jar r < i < N - 1, maxt r - .,. = maxtN-l, l\tIaxti = i = T, ..• , N- 1, and one of the following holds: a. Ft = 0. b. Ff:r t= 0 and maxw-ordN = O. c. #- 0, maxw-ordN > 0 and maxtN-1 > max I'N·

F.*

Fl- 1 1m'

6. Resolution of Singularities in Characteristic Zero

90

We will now prove Theorem 6.30. Let Eo = {Dl, ... , D s }. Let Dr be the exceptional divisor of 7l"i, where 1, = .~ + i. Set = (max w-ord 7·) > O. Let ro be the smallest integer such that maxw-ordro = maxw-ordr in (6.35).

b; b"

There exist an open cover {Wo"} of Wo and a d-dimensional structure , A' A. - A A X - >. {Eo} AEA of (Fo, Wo, Eo), where B u = (Wo , (ao' b ), Eo) for A E A. Thus for A E A, we have sequences of transformations - A A A -X (W ,(a b ), E , r r r)

where the center

Yi>.

of Wi~ 1

--,+

'll'r '11'1 --t ••• --t

- A . A A -A ,(ao, (W b ), Eo), o

T.v/ is contained in Max ti. There is an

• expreSSIon ,\

),

]

r

Ci " _ I D>O'··· -rCr -A ar .LD~ a'r •

We now define a basic object (B~)>' = (Wr\ ((a~),\ (b' ).),

on

fr:;)

W; by > - b" , if b"r < b" ,

if b"r

(a~)" =

(6.37)

if b~ > b\ if b~ < b".

Lernma 6.3l. 1. Sing((a~)\ (b' )") = (Max w-ordr ) n W;. 2. (tVr\ ((a~)\ (b' )"), E;)is a .9imple ba.9ic object.

3. A transformation

(Wr"+ 1, ((a~+l)\ (b' )"), E;+l) ~ (w,\ ((a~,)\ (b' )"), E;) with cente7' -A

Y/

induces a t'l'O/rL,sfonnat'lon

. A

A

-A

-A

A

A

-A

(W"+l,(ar+l,b ),E,.+l) --) (W,.,(ar,b ),Er )

f;." C Max w-ord,,. nWr" . Sing( (a~+ 1)" , (b')") = 0 if and only if one of the following such that the center'

4.

a. maxw-ord r > maxw-ordrt1 , or b. maxw-ord r = maxw-ordr +1 and (Maxw-ordr ) 5. Ifmaxw-ordr = maxw-ord r +1 then

Sing((a~+d\ (hI)>') = Maxw-ord r + 1 nW;+l' and (a~+l)" is defined in terms of a¢+l by (6.87).

hoLd.9:

n Wr" = 0.

91

6.7. Resolution theorems for a general basic object

W;

Proof.!. For q E we have q E Max w-ordr if and only if vq(a¢) > b~ and lIq(a~) > bA. These conditions hold if and only if vq((a~)A) > (b')A. 2 follows since vq (a;) <

b; for q E W;.

3 is immediate from 1. If b~ > bA1 we have (6.38 )

-A··~ arll = ID>.T a r

-(hi)>' ( I)A (' )>. = ID~~ ar = a r +1 .

r+l

r+J.

Suppose that b~ < bA• Then ( -A Ur

+1

)b>'.~b:

+

(6.39)

_ ('L- b;

_ )b>'-b). : DL a r r'r

-

>.) h~

>.

'Lei ... ~r+l L n >. ( D~

(~ i'TT...<¢+l) b~ L D~ .. '.L D~

r+l

where

t

,,+1

1

7'

C;+l(Q) =

L

• 1. J=

Cj (l1)VT1 (ID>.)

..,.c>'

1

b

:J

if TJ is the generic point. of the component of If we set K = .LrJ~

A

+ b; -

y;'A

containing

1rr

+l(q) .

c>'. - A .. .I;:1 (the ideal sheaf on W r ), we see that (6.39) ,+1

is equal to

(ar

(6.40)

+ K)I(b~-b~)b;: D.. r+l

-= (a~ + d A•

Now 4 and 5 follow from 1 and equations (6.38), (6.39) and (6.40). At q E Max t r (E;)A such that

n W; there exist N

= 11( q) hypersurfaces Dt 1 • • •

...' 1 n .n~n q E D~' In .1

n Max w-ordr

...i. .,..

,DtN

0 E~

0.

Thus there exists an affine neighborhood U7~,rJ> of q such that (6.41)

Max t r n UA,,p r

= D~'tP n ... n f)~'rP n Max w-ordA,tP 21 r in

(The superscript A, cf> denotes restriction to U;'¢.) Now define a basic object (B~)A,¢ =

by (6.42)

(a~}A,cf>

(b")A,cf>

(U;'¢, «a~)A,tP, (b")A,tP) , (E:)A.¢,)

6. Resolution of Singularities in Characteristic Zero

92

Lemma 6.32. 1. Singe (a,~)Alql, (b").'q,) = Max i..,. n u;',«a~~)A,eP,(b")A,q»),(Er+)A,qlJ)

is a simple basic object.

:l. A tran.'Ifonnal:ion

«a" )A,cb (b")A,cb) l (E+ (UA,fj) '1'+ l' . 'r+ l ' r+ 1 )A,4

1

)

~ (U A,41 .,.

«a")A,

l ..

with center Y;.A,cP induces a transformation A,¢ (aA,rp b~) EA,tP) ...• (U)..,cP (0,)..,<1> b)..) E)..,rP) (Ur+ l' r+ l' 'r+ 1 r' r' lJr .'J'lLch that Y/"'¢ C Max L.,.. 4. Sing(a~+l)A,cP, (b")A,cP) = ditions hold.~:

0 if and only

if one of thp. follo'wing con-

a. max t r > max t r +17 or b. maxtr = maxtr +l and Maxtr+l n u;:!i = 0. 5. If max t r

=

max t r +1 and

Max: t.l'+ 1 n W'\I f 0..

Sing«(a~+l)A,q" (ll')>.,tI»

then

= Maxw-ordr+l nu;li.

and (a~+1)A,4> is defined in terms of a;+l and (a~+l)A by (6.35) fLnd

(6. 37}. Proof. 1. For q Ear' we have q E Max t r if and only if q E Max w-ord.,. and 7J(q) = N, which hold if and only if vq((a~.)A) > (b')A (by 1 of Lemma 6.31) and l~q (IDA' 1 for 1 < j < N, which holds if and only if q E lj

Singe (a~)A,rP, (h")A,rP). 2 follows l:lince 1/q«a~)A.9'»

< (blJ)A' for q E U;,4>.

3 is immediate from L

4 and 1j follow from 4 and 5 of Lelnma 6.31, and the observation that if maxtr +l = max t r and q E Maxt"+1 n then q must be on the strict

U::0, transforms of the hypersnrfaces Dt'q" ... , Dt

tiJ

•

D

We now observe that the concJm~jons of LeUtlllaM 6.31 and 6.32, which are formulated for transformatiollil, can be naturally formulated for restrictions also. Then it follows that the (B~)A,tfi define a aBO with d-dimensional structure on (Wn (H~)+): (;::', Wn (E~)+),

(6.43)

with associated closed sets Ff'

= Max ii-

If Y is a permissible center for (6.43), then Y makes simple normal crossings with The local description (6.41) and (6.42) then implies that

E:.

93

6.7. Resolution theorems for a general basic object

Y makes simple normal crossings with E r • Thus Y c Maxt r is a permissible center for (:F!, Wr, E:). If the induced transformations are

(.1':.Hl Wr+lt (E~+l)+) -+ (.1';', Wr, (E~)+) and

(:F!rl' W r +l, E~+l) - t (:F;, Wr, E:),

F:'-rl

then either maxt" > maxt r +l, in which case = 0, or maxt r = maxtr +l and F;,+ 1 = Max t r +1 • We will now verify that the assumptions of Theorem 6.24 hold for the simple basic object. (U';,tP, ((o.~)A,tP, (b")A,¢) , (E~)A,tP). Define (by (6.37)) (B':'o)A in the same way that we defined (B:.)A. Now 5 of Lemma 6.31 implies (B:.)A is obtained by a sequence of transformations of the simple basic object (B~o)..' (E;)+ is the exceptional divisor of the product of these transformations. Since (B~O)A is a simple basic object, we can find an open cover {Vr~' '11} of W~ with non-singular hypersurfaccs (Vh);ti w in ~~j'1i such that 7

/\

.L(Vh)~oq, C u

(b')'\ -1 ( ( , ) ).., 'll )

(1,ro'

Let (Vh );''1' be the strict transform of (Vh)~O,p in Vr A,1/J. By Lemma 6.21, (Vh);'1P is non-singular,

I

.

(Vh);'tP

C ~(b')'\-l((a' )A,1/J)

r'

and we have that (Vh);'1P makes SNCs with (E:) .. ',p. Let U';,4>,1/J = v;.A,v:, n U;,rP. For fixed A and ¢, {U;,¢,1/J} is an open cover of U;,4>. For q E U;,rP,'l/J, (1(Vh);,'i.)q C

(.6(b')>'-l((a~»)""'·))q

C .6(b")>"¢-l

((a~)A,4»q

by (6.42). Thus the assumptions of Theorem 6.24 hold for (B~)A,¢ = (U7~'rP, ((a~) .. 'rP, (b"» ..'tP),

(E;:) .. ,1/J),

and {u:,~,~,,p}. Assertion 10f Theorem 6.30 now follows from Theorem 6.24 applied to our GDO of (6.43), with d-dimensional structure (B~)A,cP. If R(l )(Max t1") is empty, then by Theorem 6.24, applied to the GBO of (6,43), and itsd-dimensional structure, (6.43) has a (d - I)-dimensional structure, and we set (~-1, Wr , E:~-l)

=

(:F;, Wr , (E:)+).

Given a resolution (6.36), by Lemma 6.32, we have an induced sequence

(:F~, WN, E~)

-7 ...

-+

(F;, Wn

E~)

such that the conclusions of 2 of Theorem 6.30 hold.

94

6. Resolution of Singularities in Characteristic Zero

Lemma 6.33. Let II and 12 be totally ordered sets. Suppose that (WN, EN) ~ ... ~ (Wo, Eo)

(6.44)

is a sequence of transformations of pairstoge.ther with dose.d sett:l Fi C Wi s1J.ch that the center Yi of 1ri+l is contained in Pi for all i, 1T'i+l (Fi +1) C Fi for' all i, and FN = 0. A.9.9ume that for 0 < i < N the following conditions hold: 1. There is an upper semi-continuous function gi ; Fi

2. The1'e i.9 an 1J.pper .ljcmi-continuous function

~

11.

g~ : Maxgi ---t

12.

3. Yi = Max.Q~ in (6·44)· 4. For any q E Fi+ 1, with 0 < i < N - 2,

if 1T'i+l (q) E Yi, if 7ri+ 1 (q)¢ Yi.

gi (1ri+ 1 (q » > gi+ 1 ( q) 9i (7ri+ 1 ( q)) = 9'6+ 1( q)

This condition implies that maxgo > maxgl

~

...

~

maxgN-},

and if maxgi = maxgi+l, then 1T'i+l(Max.Qi+l) C Maxgi' 5. lfmaxgi = max9il.l and q E Maxgi+l, with 0 ~ i .~ N - 2, then gH 1ri+ 1 (q) > g~+l (q) 9~(1ri·H (q) = g'~+l (q)

if 1ri+l (q) E }i, if 7ri+ 1(q) ¢ Yi. 'Then there are upper semi-continuous functions defineAi by (6.45)

f;(q)

={

(max 9i, max gD fi+l (q)

if q E }i, if q ~ Yi,

where in the second case we c.an viem q a.'1 a point on Wi+l, a.9 Wi+l is an isomorphism in a neighborhood of q. For q E F i +1 , h( 7rHl (q» > Ii(q) Ji( 1ri+l(q) = fi+l(q)

(6.46)

maxli

if 7ri+l (q)

= (maxgi,maxgi)

E

---.

Wi

}i,

if7rHl(q) (/. Yi, and Maxh = Maxgi.

Proof. We first show, by induction on i, that the fi of (6.45) are well defined functions. Since FN = 0, YN - 1 = FN~-l' By assumption 3, Maxg',v-1 = MaxgN~-l =

FN 1,

so we can define

fN-l (q) = (9N-l(Q), 9N--l (Q») for

q E FN-l.

=

(maxgN-l, ma.xgN-l)

6.7. Resolution theorems for a general basic object

95

Suppose that q E Fr. If q E 1';., we define

fr(q) = (gr(q), g~(q))

=

(maxgr , maxg~).

Suppose that q rJ-lr,.. As 7I"i+l (Fi+l) C Fi for all i, we have an isomorphism F,. - Y;. ~ Fr +1 - Dr +l' where Dru is the reduced exceptional divisor of Wr+l ---+ Wr · Since FN = 0, we can define fr(q) = fr+l(q) for q E Fr -1';.. The properties (6.46) are immediate from the assumpt,ions a.nd (6A5). It remains to prove that fr is upper semi-continuous. We prove this by descending induction on r. Fix a: = (at, (2) E II X h. We must prove that {q E F,. I f7·(q) > o:} is closed. Ifmaxfr < £1', then the set is empty. If max fr > 0, then

{q E Fr I fr(q) > a}

= Maxg~ U 1rr +l({q' E

Fr +1 Ifr+l(q') > a:}),

which is closed by upper semi-continuity of fr+ 1 and properness of 71"1'+ 1.

0

Theorem 6.34. Fix an intege.r d > O. The.1't~ are a totally ordered set Id and functions ff with the following properties: 1. For each GBO (:Fe, Wo, Ed) with ad-dimensional .5truet'lLre and there is a functiun assuciated clused subsets

Fl fg: Fg ~ ld

with the property that M~ f8 is a permi.5sibLe center for (Wo, EX)· 2. If a sequence of transformations with permissible centers

(.r:, Wr , E:)

(6.47)

---+ •.. ---+

Yi

(:F8, Wo, E8)

Ft-·..)

and junctions fl : Id, i = 0, ... , r - 1, have been defined with the property that 1'i = Max fid, then the.re is a function f: : F~ --+ I d such that Max f!' is permissible for (Wr t E~). 3. Fur' each GBO (:F8, Wo, E3) with a d-clime.nsiunal structure, the.re is an index N so that the sequence of transformations

(6.48)

(:F8, Wo, Eg). constructed by 1 and 2 is a resolution (FN = 0). (:Ft, WN, E'fv)

---+ .•.

----t

4. The functions fid in 2 have the following properties: a. If q E Fi with () < i < N- 1, and if q rJ-}'i, then fid(q) =

n~l(q)·

b.

1'i =

Maxfid for 0 lllax fS


~ N - 1, and

> max ft > . .. > max ffv-1

c. For 0 < i < N - 1 the closed set Max ft· is smooth, equidimensionaL, and it.5 dimension i.5 determined by the vaLue of

maxf!.

6. ResolutioIl of Singularities in Characteristic Zero

96

Proof. We prove Theorem 6.34 by induction on d. First assume that d = 1. Sct h to be the disjoint union h = Q x ZU{:)()} , where
16 -

with associa.ted closed sets Fi. Definc tfi (tfi is defined by Theorern 6.28). The closed set is (locally) a codimension > 1 subset of a onedimensional subvariety of Wo, so dimFo = O. Thus R(l)(Max tA) =1= 0, and we are in the situation of 1 of Theorem 6.30. If we perform the permissible transformation with center Max t~,

FJ

(Ft, Wl~Ef) -+ (FJ, WO, FJJ), we have Inax

tb > max tl.

We can thus define a sequence of transformations

(:F;, Wr , E;)

-+ ... -+

(FJ, wO, EJ),

where each tran.e,formation has center Maxtl,

1l = t! : Fl-+ II

and

max t 01 > ... > maxtr1 · Since there is a natural number b such that maxt} E ~Z x Z for all i, t.here is an r such that the above sequenee is a resolution. Now asslllue that d > 1 and that the conclusions of Theorem ft.:~4 hold for GBOs of dinleIlsioJl d -1. Thus there are a totally ordered set 1d - 1 and 1 functions satisfying the conclusioIlB of t.he theorem.

1t-

Let 1~ be the diAjoint union

ld =

Q x Z U 1M U {oc},

where 1M is the ordered set. of Definition 6.16. We order 1d as follows: Q x Z has the lexicographic order. If (\' E Q x Z and {J E 1M, then /3 < 0, and 00 is the maximal elemcnt of 1d. Define 1d = 1d X 1d-l with the lexicographic ordering. Suppose that (.1"8, W0 1 Eff) is a GBO clolSed sets Define 90 : F o ---t 1~ by

Fl.

or dimension

d, with associated

go(q) = tg(q),

and 9b : Max90

-+

Id-1

by

g~(q) = { ~-l(q) wllere

if q E R(l)(Max tg), if q ¢ R(l)(M~tg),

18-1 is the function defined by the GBO of dimension d ( J"C"d-l 0

of 2 of Theorem 6.30.

UI ,I'I' 0,

ed-I) .:10

1

6.7.

Re1:;olutioll t11eorems for 8. general basic object

97

Assume t,h8ot we have now inductively defined a sequence of transformations

(6.49) and we have defined functions .Qi :

Ff ~ I d and g~ : Maxgi ~ I d - 1

satisfying the assumptions 1-5 of Lemmo. 6.33 (with II = I d and 12 = I d for i = 0: ... ) r - 1. In particular, the cent.er of 7l'i+l iE "Yi = MaxO~. If F:l =1=

0,

define Or : F;

-----+

1)

I dby

( ) _ {r(Dr)(q) if w-ord~(q) = 0, .Qr q t~(q) if w-ord~(q) > 0, where f(B r ) is the function defined in Proposition 6.29 for the GBO B r = (:F.~, Wr, E:). Define g~ : Max 91' -----+ I d - 1 by

where

if w-ord~(q) = 0, if q E R(l)(Maxt~) and w-ordr(q) if q ~ R(l)(Maxt~) and w-ord.,.(q)

9~(q)

=

f;-1

is the function defined by the GBO of dimension d - 1

> 0, > 0,

~-1 W b,d-l) ( .r;: , 1', r

in 2 of Theorem 6.30. Observe that if maxw-ord r > 0, then t.he ~enter Yi of 7ri+1 in (£1.49) is defined by 1'i = Maxgi c Maxtf for 0 < i < r - 1. Thu~ (if maxw-ordr > 0) we have t.hat «(i,49) is a sequence of the form of (6.35) of Theorem 6.30. Now Theorem fi.;·m, our induction assumption, and Lemma 6.17 show that the sequence (6.49) extends uniquely to fl resolution

(TN, WN, EN) ~ ... ~ (To? W u, Eo) with fundiolls 9i and g~ satisfying a..~sumptions 1-5 of Lemma 6.33. Lemma 6.33, the functions (gi, gD extend to fundions

Fi ~ ld ~ l~ x d

By

Jd-l

satisfying the conclusions of Theorem 6.34.

ft

Consider the values of the functions which we constructed at a point q E Fi. fid(q) has d coordinates, and has one of the full.owillg Lhree t.ypes: 1. fid(q) = (td(q), td-1(q), 2. fid(q) = (td(q), t d l(q),

3. fid(q) = (td(q), t d - I (q),

tdr(q), 00,·" ,00). , t d- 1'(q), I'(q), 00,' .. ,00). , t 1 (q)). l

98

. 6. Resolution of Singula.ritie.s in Cllaracteristic Zero

Each coordinate is a function deB ned for a GBO of t.he corresponding dimension. We have t 1(q) = (w-ordi(q), T}i(q)) with w-ordi(q) > O. In case 1, t d - r is such that q E R(l)(M.~fl-r)j that is, Max t d -'" ha.~ r:ndirnensinn 1 in a. d - l' dimenl=lioual GBO, and dim (1Iax fid ) = d - r - 1. III case 2, w_ordd-~"'+l)(q) = 0 and r is the function defined in Proposition 6.29 for a monomial GBO and dim (Max ft) = d - r - rl -1, where rt is the first coordinate of max r. In case 3, dim(Max f id ) = O. Thus ~1~ IiI is non-singular and equidinlfmsinnal, and 4 c lallows.

0

Theorem. 6.35. Suppose that d is a positive integer. Then there are a totally ordered set I d and functions with the following properties:

pf

1. For each pair (Wo, Eo) with d ('?wo there is

= dim Wo

and

t:~a.ch

'ideal sheaj Jo

C

a function 1/g: V(Jo) -+ I d

with the pT'opeT'ty that 1Iaxp8 c V(Ju) is permissible for CWo, Eo). 2. If a sequence of transformations of 1/air8 with Cf:nlt:Ts Vi. C V(J·d

(lV1' , E r )

-+ ... -+ (Wu,

pf

Eo)

and functions :V(Ji) -- I d, lor 0 < i < r - 1, has been defined ·w'i./.h I.he p1'ope'd.y that Yi = ~1ax and V (J k) :1= 0, then there is a junction P: : V (J r) .• l such that M~p~ is l1crm:i.9.9iblc for

I:

pf

( W", FJ1.).

3. For each pair (Wo, Eo) and Jo c that thf:

8eq1J,ent~f'.

O",°u

there is an index N such

oj /'1'(L'rUtj(}'rrnlLl'tons

(WN,EN) -+ ... -- (~VolEo)

constructed inductively by 1 and 2 i.e; such that V (J N) = 0. 1 'h.t~ COTTf:8ponding 0'4eq'll.t-:1I.Cf': will be called a UstT'Ong pTincipalization" of Jo· 4. P7'operty 4 of 1'lu~orf:1n. 6.84 holds. Proof. Let Jo = Jo, and set bo - max 11-./ • ny Theorem 6.34, there exists , 0 a resolution (WNl1 (JNu bo), E Nl ) --+ ... --+ (Wo,

(Jo, boL Eo)

of the simple basic object (Wo, (Jo, bo), Eo). Thus we have that .1 Nl (which is the weak traJ1sform of .10 on OWN) satisfies b1 - ma.x V ·J < boo We now 1 Nl can construct by Theorem 6.34 a resolution (WN2' (.1 N2~ b,), EN.J -- ... -- (WN" (J N" b1), EN1)

99

6.8. Resolution of singularities in cllaracteri"!tic zero

of the simple basic objeet (WNu (JNl,bI),ENl)'

Aft.er constructing a finite number of relSolutions of simple basic objects in this way, we have a. sequence of transformations of pairs (6.50) such that the strict transform J N of .10 on W N is J N =

OWN'

We have upper semi-continuous functions

ft : Max VJ

i -

lei

satisfying the condusions of Theorem 6.34 on the resolution sequences

(WNnu (J N H1 , bi), ENi+l) ............. (WNp (J N .. , l)i),HNJ. Now apply Lenuna. 6.33 to the sequence (6.50), with Fi = V(J'i), gi L'J.1 g~ = We conclude that there exist runctionspf ; V(Ji) ------+ I d = Zxld with the desired properties. 0 ~

if.

6.8. Resolution of singularities in characteristic zero RecAll our conventions on varieties in the notation part of Section 1.1.

Theorem 6.36 (PrincipaJiza.t.ion of Ideals). Suppose that I is an ideal sheaf on a non-singular variety W over a field of characteristic zero. Then there exists CJ, sequence of monodial transforms 7r:

WI ..... W

which is an isomorphism away from the closed locus of points where I isnol locally principal, such that IOWl is locally p1'incipal. Proof. We can factor I =. I1I21 where II is all invertible sheaf and V(I2) is the set of points where I is not locally principal. Then we apply Theorem 6.35 to Jo = I2. 0 Theoreul 6.37 (Embedded Resolution of Singularities). Suppose thai, X i,q an algebraic variety over a field of chamcteri9tic zero which is embedded in a non-.',ing'lLlar variety W. Then there exists a birational projective morphism 7r:

WI ..... W

such that 7r is a sequence of monodial transforms, 7r is an embedded resolution of X, and 7r is an isomorph-ism a'way from, the singular locus of W. Proof. Set X o = X, Wo = W, J o = Ixo c OWo' Lp.t

(WN, rJN)

--4

(Wo, Eo)

100

6. Resolution of Singularities in Characteristic Zero

be the strong principalization of Jo constructed in Theorem 6.35, so that the weak transform of .10 on WN is .1 N = OWN' With the notation.ofthe proofs of Theorem 6.34 and Theorem 6.35, if Xo has codimension r in the d-dimensional variety Wo, then for q E Reg(Xo),

pg(q) =

(1, (1, 0), ... , (1, 0), 0

I ••• ,

0)

E I~,

where there are .,. copies of (1,0) followed by d - .,. :teras. The function

pg

is thus constant on the non-empty open set Reg(Xo) of non-sin~lar points of Xo. Let this constant be r. E 1:1, By property 4 of Theorem 6.84 there exists a unique index r < N - 1 such that max p~ = c. Since Wr --+ Wo is an isomorphism over the dense open set Reg (Xo), the strict transform X r of X 0 IIlust be the union of the irreducible cornpOIlents of the closed set M a.x p~ of W r . Since Maxp~ i,s a permissible center, X r is non-singular and makes silnple normal crossings with the exceptional divisor E r • D

Theorem 6.38 (Resolution of Singularities). Suppose that X is an algebraic 'variety over' a field of characteristic zero. Then the'T'e is lL resolution of singularities 7r:X 1 ---+X

such that 7r i.CJ a projective 1norphi,c;1n which i.CJ an i,c;omorphism away from the singular locus of X. Proof. This is immediate from Theorem 6.37, after choosing an embedding of X into a projective space W. D Theorem 6.39 (Resolution of Indeterminacy). Suppose that K is a field of r.haraeteri.CJtir. ZC1'O and (/) : W ~ V i.., (L rat'ion(Ll map of projective Kvarieties. Then there exist a projective birational morphism 71" : H'l ~ W such that WI is non-singular and a morphism A : WI ---+ V such that A = ¢ 0 1r. If W i.s non-singular', then 7r i8 a IJ'rodw:t of 111.ono'ida.l transform.'i. Proof. By TheoreIll 6.38, there existlS a resolution of ~ingularities 't/) : WI ---+ W. After replacing W with WI and ¢ with 1/J 0 4>, we may assume that W is non-singular. Let r q, be the graph of ~ D(I) is the blow-up of I (Theorem 4.5). By Theorem 6.36, there exists a pl'incipalization 7r : Wl .~ W of I. The universal property of blowing up (Theorem 4.2) now shows that there is a morphism A: WI --+ V such that A =

e

Theorem 6.40. Suppose that 7f : Y ---+ X is a birational morphism of projective non-t~ing?llar 1JarietiB.'3 over (J, field K of eharar.teri.CJtic O. Then

Ili(y, Oy)

Co!

IIi (X, Ox)

'V i.

6.8. Resolution of singularitie8 in characteristic zero

101

Proof. By Theorem 6.39, there exists a projective morphism f : Z ---+ Y such that 9 - 7r 0 f is a product of blow-ups of non-singular subvarieties,

g: Z - Z~n

9n ---+

Z n-l

92 Z ---+ •.• ---+ ·1

9n-1

91

-_.~

Z0

= X.

We have ([65] or Lemma 2.1 [31])

Ri 9j* 0 zJ

=

{.. 0, 0

i> 0, Zj-1 ,

i.= O.

Thus, by the Leray spectral sequence,

(6.51)

Rigtl
=

{ci •. , r

}

if'i if i

> 0, = 0,

and

gtl< : 111(X, CJ x)

rv

Hi(Z, Oz)

for all i. Since g* = f* 07[* I we conclude that rr* is one-to-one. 'lb show that 7T* is an isomorphism we now only need to show that. f* is also one-to-one. Theorem 6.40 also gives a projective morphism 'Y : W ---+ Z such thaI, f3 = f 0 "y is a product of blow-ups of non-singular subvarieties, so we have

/3* : Hi(y, Oy)

~ H'i(W, Ow)

for all i. This implies that f* is one-to-one, and the theorem is proved.

0

For the following theorem, which is stronger tha.n Theorem 6.38, we give a complete proof only the case of a hypersurface. In the embedded resolutiun constructed in 'I'heorem 6.37, which induces the resolut.ion of 6.38, if X is not a hypersurface, there may be monodial t.rallsfonns Wi + 1 ---+ Wi whose centers are not contained in the strict transform Xi of X, so that the indl.leed morphism X i +1 ---+Xi of strict transforms of X will in general not be a monodial transform.

Theorem 6.41 (A Stronger Theorem of Resolution of Singularitie~). 87/.1'pose that X is an algeln'(Lic variety over a field of characie1"i"'If,ic ze.ro. Then there is a re.-ml'l£t'ton of singularities 7t' :

Xl

---+

,X

such that 7r is a .sequencf:'; of monodial transforms cente7'ed in the. dosed sets of point.'l of maximum multiplicity.

Proof. In the case of a hypersurface X = Xo clnbedded in a non-singular variety W = Wo, the proof given for TheoreIIl 6.37 actually produces a resolution of the kind asserted by this theorem, since the resolution sequence is constructed by patching together resolutions of the simple basic objects (Wi" (Ji , b), Ei ), where Ji is the weak transform of the ideal sheaf Jo of Xo in Woo Since Jo is a locally principal ideal, and each tran..~formation in

a. R.esolution of SinfrUlarities in Cha.ra.ct.eristic Zm'o

102

these sequences is centered at a non-sing-ular subvariety of Max.J i, the weak transform J i is actually the st.rid transform of Jo. For the case of a general variety X, we choose an elnbedding of X in a projective spaee W. We then modify the above proof by considering the Hilbert-Samuel function of X. It. eRn be shown that there is a simple basic ohject (Wo, (1(0, b), Eo) such that Max vKo is the lllaximal locus of the Hilbert-Sarlluel runction. In the construction of of Theorem 6.35 we use 1/[(0 in place of lI"Ji ' Then the resolut,ion of this basic object induces a sequeuee of monodial transformations of X such tha.t the maximum of the Hilbert-Samuel function haJj dropped.

pf

More details about this process are given, for instance, in [55], [14J and [7~.

0

Exercise 6.42. 1. Folluw the algorithm of this chapter to construct an elnbeddcd resolution of singularities of: a. The singular curve y2 - x a = 0 in A 2 . b. The singular surfacez 2 - x 2y 3 = 0 in A3. c. The singular curve z ~ 0, y2 - .-x 3 = 0 in A3 .

2. Give examples where the Theorem 6.31.

fl

achieve the 3 ccu;es ul" the proof of

3. Suppose that. K is a field of characteristic zeru allel X is an integral finite type I(-schelne. Prove that there exists a proper biratiolml morphism 1f : Y ----; X such that Y is Hon-singular. 4. Suppose that K i~ a rield of characteristic zero and X is a reduced finite type K -scheme. Prove that there exists a proper birational morphi:-;rn 7r : Y ~ X such that Y is non-singular.

5. Let notation be as in the statement of' Theorem 0.35. a. Suppose that q E: V(Jo) c Wo is a closed point. Let U = spec(OwO,q), D i = D·inlJ for 1 < i < rand E = {D 1l •· . l Dr}. Consider the function of Theorem 6.35. Show that pg(q)

pg

depends only Oll the local basic object (U, E) and the ideal (Jo)q c OWU,q. In particular, pg(q) is independent of the Kalgebra isomorphism of OX,q which takes JoVX,q (;0 .JoVX,q and Di to Di for 1 < i < r. b. Suppose that H : ltVo .~ Wo is a K -automorphism such that 8*(Jo) = Jo. Show that pg(8(q)) = p3(Q) for all q E V(Jo). c. Suppose that 8 : Wo ~ Wo is a K-automorphism, and Y c Wo is a non-singular subvariety such that 8(Y) = Y. Let 7rl : Wi ~ W be the mOlloidal transfonn centered at Y. Show t.ha,t. extends to a K-automorphism WI ~ W 1 such that

e

e:

6.8. Resolution of singuJaHties in characteristic zero S(1rt(q) = 11"1 (8((1) for all q E WI and SeD) - D if D is the exeeptional divisor Of7Tl' 6. Suppose that (Wo, Eo) is a pair and Jo c OWo is an ideal sheaf, with not.ation a.g in Theorem 6.35. Consider the sequence in 2 of Theorem 6.35. Suppose that G is a group ofK -au LOUlurphisIns of Wo such that S(Di) = Vi for 1 < i < rand S*(Jo) =Jo. Show that G extends to a group of K-automorphisms of each Wi. in tJlP. sequence 2 of Theorem 6.35 such that e(}'~i) = Yi for all 8E G and each 7fi is G-equivarianf.. 'Tha.t is,

7Ti(8(q) = S( 7Ti(q)) for all q E Wi- Furthermore, the functions pi of Theorelll 6.3[, satiefy pf(8(q» -= pt(q) for all q E: Wi· 7. (Equivariant. resolutiun of singularities) Suppose that X is a variety over a field 1< of characteristic zero. Suppose that. G is a. gruup of' K -automorphisms of X. Show tha.t there exists an equivariant resolution of singularities 7r : Xl ~ ,,'l( of X. That is, G extendR to a group of K-automorphisms of X so that 11"(8«(:1) = 8(7r(q» for all q E Xl. l\lore genera.lly, deduce equivariant versions of all the ot.her t.heorems of Section 6.8, and of parts 3 and 4 of Exercise

6.42).

~

Chap'ter 7

Resolution of Surfaces in Positive Characteristic

7.1. Resolution and

SOllle

invariants

In this chapter we prove resolul.ion of gingularities for surfaces over an algebraically closed field of characteristic p > O. The characteristic 0 proof in Sectioll 5.1 depended on the existence of hypersurfaces of IIlaximal contact, which is false in positive eharaet.eristic (sec Exercise 7.35 at the end of this section). The algorithm of this chapter is sketched by Hironaka in his notes [53]. We prove the following theorem:

Theorem 7.1. Suppose that S is a surface over an LLlgelJ1'(ji(~ally elOllcd ficld K of characterist,tC P > O. Then therc cXists a resolution of singularitie.fj

A :T

-+-

S.

Theorem 7.1 in the case when S is a hypersurface follows from induction on r .= max{t I Singt(S) 1= O} in Theorems 7.6, 7.7 and 7.8. We then obtain Theorem 7.1 in general from Theorem 4.9 and 1'}lf~orem 4.11. For the reIIlainder of t.his chapter we will assume that K i~ an algebraically closed field of characteristic p > 0, V is a non-singular 3-dimensional variety over K, and S is a surface in V.

It follows from TlLeoreUI A.19 that for t t- N, the set Singt(S) = {p E S I vp(S)

> t}

is Zariski closed.

105

7. R.esolution of Sllrfaces in Positive Glmracl:erisl,;c

106

Let

'T' =- max{/, I Singt(S)

#- 0}. is a dosed point, f =

Suppose that 11 E Singr(S) 0 if:) a local equation of " S at p, and (:1;, y, z) are regular parameter~ at p in V (or in ('Jy',p). Tllere is • an expansIOn . k f = 'L" ' aijk Xi ylz i+i+k>r

with

(J,'i,jk E

K in

"

OV,p -

Kr[~z;,

y, zJ1. The leading forrn of

L(x, y, z) =

'"' L

f

is defined to be

. . k.

aijkxlyJ z

i+i+k=r

We define a new invariant., T(P), to be the dimension of the smallest linear subspace M of the !{-subspace spanned by x, y and z in K[x, y, z] such that L E: k[lVIJ. This subspace is uniqnely determined. This dimerlsioll is in rad independent of the choice of regular parameters (:z:, y, z) at p (or in CJY,lJ ) ' If ;1:: y, z are l'f~glJlal' !larameLers in OV,Pl we will call the subvariety N = V(M) of spec (Ov,p) an "approximate manifold" to S at p. If (x, y, z) are regular parameters in Ov,p, we call N = V(M) c spec(Oy,p) a (formal) approximatc manifold to S at ]J, AJ is dependent. on our choice of ,'egular parameters at p. Observe that ~

1
< 3.

lfthere is a non-singular curve C c SingT.(S) such that q E C, then r(q) < 2. Example 7.2. Let f = ;];2 + y2 + z2 + :1: 5 . If ch8or(K) f 2, then r = 3 and ~D = Y = z = 0 are local equations of the approximat.e manifold at the orip;in.

However, if char(K) = 2, thca T = 1 and x of an appl'oximate manirold 801. the origin.

Example 7.3. Let

+y +z = 0 is a local equation

.f .= 1;2 + 2xy + x 2 + z2 + z\ char(K) > 2. Thcn L = (y + x)2

+ Z2,

so that T =2 and x + y = z = 0 are local equat.ions or an approxima.t.e manifold at. t.he origin.

Lemma 7.4. Suppose that q E ..9 is a closed point .
7.1. Resolution and some invariants

Proof. Let

1=

107

0 be a local equation of S at lJ, and let (x, y, z) be regular

parameters at p such that 1.

2.

= y = z = 0 are local equations of N if r(q) = Y = z = 0 are local equations of N if T(q) = 2,

;r,

3,

3. Z = 0 is a local equation of N if T(q) = 1.

In Ov,p = K[[x, y, z]], write ~ i' k f = L.; (Lijk:r, 11 z = L + G,

where L is the leading form of degree 1', and G has order> r. ql has regular parameters (Xl, Yl, Zl) such that one of the following eases holds:

= Xl,Y = Xl(Yl + a),z = Xl(Zl + (3) with et,(3 E K. 2. x = XIYI, Y -= Yl, Z ~ Yl (Zl + (I) with lJ E= K. :i. x = XIZl,Y = YlZl,Z = Zl. 1. x

Suppose t.hat the first case holds. 8 1 has a local equation such that It = [J(l,YI +~,Zl +/j) +xl!"l

Thus

lIq ]

(11)

< r, and

lIq1

L(l,Y1 +a,zl +~)

(!l)

Uijk(Yl

+ u}i(zj +(I)k =

bjk E

L

bjky{zf

i+k=1'

i+j+k=r

for SOTIIe

= 0 at ql

= r implies

L

=

!I

k. Thus, t.he equalit.y

L(l, Y, z) =

x x

Lk

j+ =r

(Y - H)-i(z -lJ)k X

X

implies V(y-ax, z-;3x) c N. We conclude thatr(q) < 2 if l1q1 (Sd - Vq(S). Suppose t.hat T(q) -, 2 and lIq1 (8,) = lIq (S). Then we must have a =

13 = 0, so that

ql

E N l . We further have

!l = L(Y1, Zl) + Xln, so that r(ql)

/3

> r(q).

If r(q) = 1, then L = cz'" for some (non-zero) c E K, and we must have = 0, so that ql E Nl. We further have

it = L(zl) + x 1 !l, 80

t.hat

T(qI) > T(q).

There is a similar analysis in eases 2 and 3.

D

Lemma 7.5. Suppose that C c Singr(S) is a non-singular curve and q E C V, .~ V be the blow-up of C, E = 11"-1 (C) the exceptional divisor, and 8 1 the strict transfo17n of S on VI. Suppose that

is a closed point. Lei.

'Tr :

108

7. H.esolutioIJ of Surfaces in Positive Characteristic

(x~ y, z) are regular ]J(J.T'ameter·s in OV;q (or OV,q) such that x = y = 0 are local equations of C at q, and x = 0 or x = y = () are local equations of a

(possible formal) approximate manifold N of S at q. Let N l be the strict tran.(jform of N on Vi. Suppose that ql E If-l(q). Then lIq1 (Sl) < r, and if V q1 (Sd = r, then r(q) < r(ql) and ql E N 1 n E. Furthermore, Nl n E = 0 if T(q) = 2, and N 1 n E is a curve which maps isomorphically onto spec(Oa,q) (or spec(c?c,q) in the case where N is formal) if T(q) = 1. We omit the proof of LClnlna 7.5, as it is similar to LeIIllna 7.1.

Theorem 7.6. There exists a sequence of blow-ups of points in Singr (8J. Vn~'·) Vn- t ~ • • • -+ V,

wher'e Si is the strict tronsform of S on lti, so that all are non-singular.

CUr1JC.~ in Singr(Sn)

Proof. This is possible by Corollary 4.4 and since (by Lemma 7.4) the blow-up of a point in Singr (8i ) CRn introduce at most one new curve into Singr (8i+d, which must be non-singular. D

Theorem 7.7. Suppose that all CUr1Je.'I -in Si ng r (S) are non-singular, and . . . ---+

Vn

---+

Vn -

l ---+ • • • ---+

V

is a sequence of blow-ups of non-singular C1L7'7Jes in Singr(Si) where Si is the strict transform of 8 on Vi· Then Sing,.(Sd is a union of non-singular c'Ur"ues and a finite number of points for all i. Further, this .geq1J.enCe i8 Jinilf:. That is, there exists an n such that Sil1gr (8.ll,) i.f4 (L Jinite set. Proof. If 11' : Vi ~ V is the blow-up of the non-singular curve 0 inS i ngr ( S') , and C1 c Singr(St) n 7r- 1 (C) is f1 curve, ~hell by Lemma 7.5, C1 is nonsingular and maps isoIIlorphically onto C. Furthermore, the strict transform of a non-singular curve must be non-singular, so SingrU'·i.) I1lI1St. be a uuion of non-singular curves and a finite number of points for all i. SUPPol=le that we can construct an infinite sequence . . . -+ ~t

---+

l!;l

1 ---+ • • • ~

V

by blowing up non-singular curves in Singr(Si). Then t.here exist curves Ci c Singr(Vi) for all i such that o.i+l ---+ Ci are dominant morphisms for all i, and

1I0Vi'C~. (Isi,cJ = r for all i. We have an infinite sequence

(7.1)

Ov"c

.~

Ov;1.1 c

---+... ---+

Ov; c n, 'l"

---+ .••

109

7.2. r(q) = 2

of local rings, where~ after possibly reindexing to rem.ove the rnap8 which are the identity, each homomorphism is a localization of the blow-up of the maximal ideal of a 2-dimensional regular local ring. As explained in the characteristic l.ero prool' of resolution of surface singularities (Lemma 5.10)~ we can view the Q8i,ci as local rings of a closed point of a curve on a nonsingular surface defined over a field of tran..'1cendence degree lover K. By TbeoreIIl 3.15 (or Corollary 4.4), 1I0Vl~,cTl (I.5n,cn)


o

for large n, a contradiction. We can now state the main resolution theorem.

Theorem 7.8. SnppoRe that Singr (8) 'i.s a finite set. Consider' a seque.nce. of blow-ups

(7.2)

...

where V'i+l and

Vi+1

--t

\!.;t

--t •.• --t

V2

--t

VI

--t

V,

Vi is the blow-up of a curve in Singr(Si) if such a curve exists, Vi i.~ the blow-up of a point in Singr(H·i ) otherwise. Let Si be

--t

--t

the strict transform of S on V;. Then Sing.,.(Si) is a union of non-singular curves and a finite number of points for all i. Further, this sequence ic; finite. That ';,s, the.1'f~ e:C;'!::Its \1;1, .'w,ch that Singr(Sn) = 0.

Theorem 7.8 is an immediat.e consequence of' Theorem 7.9 below, and Theorems 7.6 and 7.7. Theorem 7.9. Let the assumptions be as in the statement of Theorem 7.8, and suppose that q E Singr (S) is a closed point. Then there exist.,; an n ,,;uch tha.t. a.ll dosed point,r; qn E 8 n snch that l]u maps to lJ and Ilqn (Sn) = r satisfy T(qn) > r( q).

The proof of this theorem is very simple if r(q) = 3. By Lemma 7.4 the multiplicity nlust drop at all points above q in the blow-up of q. We will prove Theorem 7.9 in the ease r(q) = 2 in Section 7.2 and in the case r(q) = 1 in Section 7.3.

7.2. r(q) = 2 In this section we prove Theorem 7.9 in the case that T (q) = 2. Suppose that r(q) = 2, and that there exists a sequence (7.2) of infinite length. We can then choose an infinite sequence of closed points qi E Singr(Si) such that qi+l nlaps to q.i for all i. By Lemma 7.4, we have T(lJi) =2 ror all i, and each qi is hiolated in Sing1.(Si). Thus each map Vi+l --t Vi is either the blow-up of the unique point q.i E Sill~(8i) which maps to q, or it is an isomorphism over q. After

110

7. Resolution of Surfaces in Positive Gl1aracteristic

reindexing, we may assume that each map is the blow-up of a point qi such " that q.i+l maps to qi for all i. Let Ri = O\li,qi. We have a sequence

Ro---. R 1

(7.3)

---+ ••• ~

Ri

-4 •..

of infinite length, where each Ri+l is the completion of a local ring of a closed point of the blow-up of the maximal ideal of spec(Ri), and vqi (Si) = rand T(qi) = 2 for all i. We will derive a contradiction. Since r(q) = 2, there exist regular parameters (x, y, z) in Ro such that a local equation f = 0 of S in Ru has the forIlI

and the leading form of f is

L

L( :li, y) =

aijoa;lyJ.

i+j=7'

For an arhitrary series 9 -= ~ biikxiyJ zk E ~,

define -Yoyz (g)

= min{

t ") I

b;jk

r- 'l+J

i- () and i -I- j

< r}

E

-.!., N U {oo}

r.

with the conventioJl tha.t "Y:l:YZ (g) = 00 if and only if bi;ik = 0 whenever i + j < r. We have '1xyz(g) < 1 if and only if VRo(g) < ".. f<\lrthermore, 'Y:z:yz (g) = 00 if and only if 9 E (x, y) r. Set '1 = '1xyz (09 ). Define

(7.4)

L

=

[g]xyz

1JiJk:1:iyi zA~.

(i+jh+k""'-r1'

There is an expansion

(7.5)

9 = [g]xyz

2:

+ (i+ j

Returning to our series and

T.., = ((i,j)

Ir

-

h+k >T"'f

I, which is a local equation of S, set 'Y = 7xyz(!),

t+ ") = t

bijkxiyi zk.

.1

-y, i + j < ", for some k silch that

aijk

i- OJ.

/3 E

]( such

Then (7.6)

[I] xyz -- I

J

+

~

L....,

a

i,j,,(r-+~

j)

x i .. J ,..: y(r-i-j) IF N •

(i,j)E1~

Definition 7.10. [J]xyz is .rtOl7Ja,hle if 'Y E N and there exists that [f]xyz = L(~z: - o:z', y- (Jz').

0:,

7.2. r(q) = 2

111

Lemma 7.11. There exists a change of variables n

=- J: -

J:l

L

n

(l'.i Zi ,

L {.-JiZ

YI = Y -

i=l

such that

i

i=l

[f]xll z is not solvable.

H.emark 7.12. We can always r.hoose (:1:, y, z) that are regular pa.rameters in OVO,qo' Then (x, y, z) are also regular parameters in OVO,qo' However, since we are only blowing up maximal ideals in (7.2), we may work with any formal system

or regula,f' pararnet.ers

(:l:, y, z) ill R{J (see Lemma 5.3).

Proof. We prove 7.11. IT t.here rloesn't. exist. l'ltJch fl, change or variH.bles, then there .exists an infinite sequence of changes of variables for n E N n

Xn

= :1;-

L O,iZi, •

Yn - L.Bi Zi' i=-l

i-I

such that "'In E N for all n, '"'In+l - '"'IxnYn Z(!), and ')'n+l > '"'In for all n. Let 00

00

X oo

=

x-

L

o'iZ "Yl ,

L !3iz

Yoc = Y -

"Yl •

i-I

i-I

Thus

and

00

!

E

i

(x - L Cti ZZ , Y - L.8i Z · i=l

But by construction uf

00

t

c 1l4.J.

i-I

va, p iJS isolated ill Sing (8). If I c C'Jv,q ilS the reduced r

ideal defining Sing.r(S) at q, then by Remark A.21, j = I Ro is the reduced ideal whose support is the locus where f has multiplicity T in spec(Ro). Thus (x,y,z) = j c (x- L:CtiZi,y-~.8iZi),which is impossible. 0 We may thus assume that [f]xyz is not solvable. Since the leading form of f is L(x,y) and r(p) = 2, RI has regular parameters (Xl.Yl, ZI) defined by

(7.7) Let

x = x, Z 1 ,

II = !r .. = 0 be a

Y = y, Zl ,

Z

=

ZI•

local equation of the strict transform of

f

in Rl .

Lemma 7.13. 1. '"'IxIYIZ(fI) = '"'Ixyz(f) - 1. 2. [!I]Xl1,11 Z = zlr [f]:rYZ1 so Ulxyz not solvable implies [/I]:ryz not solvable.

3. If'"YX1YP(fd > 1, lhenlheleadingfo1'7nofft i.sD(:r.l,Yl) = ;rL(J:,y).

7. Resolution of Surfaces in Positive Characterjstic

112

Proof. The lemma follows from substitution of (7.7) in (7.4), (7.5) and

(7.6).

0

We claim that the case "YXl'Yl'Z(.f1) = 1 cannot occur (with our a..'1sumption that lIql (81) = 1" and T(qI) = 2). Suppose that it does. Then the leading form of It is [h]Xl1}t Z , and there exist a form W of degree rand a, b, c, d, €, f E K such that

[/1]:L:} Yl'~ = \Ji (aXl ~'e

+ bYI + CZ, dXl + eYI + f z ) .

have an expression

so that

= \II (ax + by, dx + ey).

L(x, y)

¥ 0, since T(q) =

2. lly Lemma 7.14 below~ solvable, a contradiction to 2 of Lemma 7.1:"1.

'\\Te have ae - bd

[hL£1YF is thus

Since lX1Y1 Z (!l) < 1 implies Jllll (fl) < T, it now follows from Len1ma 7.13 that after a finite sequence of blow-ups Ro ~ R i , where R i has regular parameters (Xi, Yi, Zi) with , ,,. -

••'

'J'.

..,'t

zti,

Y -- 'y'Z -t ,,~ ,

':' IV

-

".,

""'1'

By 3 of LemnlR 7.1:i, we reach a reduction llqi (8,) < 'f' or v lJi (Hi) = r, r(qd = 3, a contradictioIl too our a..'1smnption that (7.3) has infinite lent:,rtl.. Thus Theorem 7.9 has been proven when T(q) = 2.

Lemma 7.14. Let q. and suppose that

= ep(x,yL

\II = \II(u, v} be f01 ms uf u

_",U'

~(X, y)

dP.gTP.P. T

over /(

_

= \II (ax + bYI dx + ey)

for :wme a, b, d, e E ]( with ae - bd =1= O. Then fm' all r., f E K there exist Q', ,3 E ]i such that (x -I az, 'Y + /32') = \II(ax + by + cZ , dx:

+ ey + Iz).

Proof. Indeed,

q.(x + OZ, Y + /3z) ,..

W(a(x + uz) + bey + jJz), 'd(x +Q'z) + e(y + fiz» \l1(ax + by ,I· (Cia + bl3)z, dx + e.y + (do + ~-Ij)Z)l

so we need only solve

for

a, J~.

0

7.3. ;(q) = 1

113

7.3. T(q) = 1 In this section we prove Theorern 7.9 in the ca.e,e that T(q) = 1. We now a~Fmme that r(q) = 1. Suppose that there is a sequence (7.2) which does not terminate. We can then choose a sequence of points qn E Vn, with qo = q, such that qn+l maps to qn for all nand vqn (8n ) = 1', T(qn) = 1 for all n. A ('tel' pO~Ribly reindexing, we may assume that for all 'n, qn ,- is on the subvariety of Vn which is blown up unrler\l;1.+1 --> Vn . Let R n = OV ,Qn for n > n. We then have an infinite sequence I1

R=

(7.8)

~ ---+

Rl

---+ ••• ---+

Rn

.... ~ ....

Let. Ji C 0Vi,l/i be the reduced ideal defining Sing",(Si) at qi. Tlien by Lemmas 7.4 and 7.5) Ji is either t.he maxhllal ideal mi of OVi,q.j or a regular height 2 prime ideal Pi in OVi,qi' Then the multiplicity r locus of 6s.i ,q., (whidl i~ a quotient ofR;, by a principal ideal) is defined by the reduc~d ideal .h = ·JiRi by Remark A.21. We have Ji = rn.i = 'ffl"iRi or ·h_ = Pi = ]JiRi, a regular prime in R.i. By the construction of the sequence (7.2), we ~ee UlaJ OVi+l,qt+l is a local ring of the blow-up of m'i if .Ii = mi, and a local ring of t.he blow-up of Pi otherwise. Thus Ri + 1 is the completion of a local ring of the blow-up of i or Pi. ~

~

~

.,h

Vve are thus free to work with formal pal'ameter~ and equations (which define the ideal ISi ,q; = Ts; ,qi H,i) in the R;" since the ideals r11i and Pi are determined in Ri by the umlUplicy l' locus of Sing,. (6s."qJ. Suppose that T = ]([[x, y, zlJ is a power series ring, 'f' E Nand ,9 =

L bijk:L:i'!l

zk E

T.

We call construct a polygon in the following way. Define ~

A = u(,9;x,y)z) =

{(

. ·z k'

r-

.

J k) E Q21 k

r-

< rand

bijk

10}.

Let I~I be the smallest convex set in R2 sueh t.hat. ~ C. I~I and (a, b) E I~I iUlplie~ (n + c, b + d) E:-; I~I for all c, d > O. For a E lR, let 8(a,) be the linc through (a,O) with 810pe -1, and V(a) the vertical line throug-h (a, 0). Suppose that I~I =f 0. We define O'Xyz(g) to he the ~mallest a appearing in any (u" b) E I~ I, and /~XllZ (g) to be the smallest b such that

(a xyz (g), b) E I~ I· Let "Yxyz(g) be the smallest. number, such that S(,)nl~1 =f 0 and let 6xyz (g) be such that ("Yxyz(g)-6xyzeq),6xyz(g)) is the lowest. point on S'(')'xyz(g ))nl~l. Then (axyz(g),/Jxyz(g)) and (')'xyz(g) - c5xy,:Cq),6xyz(g)) are vert.iees of I~I.

7. Ret:)olution of Surfaces in Positive Characteristic

114

Define f.xyz(g) t.o he tohe absolute V'etlue of the largest slope of a line throup;h (ct xyz (g), ;3xyz (g)) such that no points of I~ (g; x, y, z) I lie below it.

Lemma 7.15. 1. The vertices of 16.1 are points of 6., which lie on thf. lattice ~! Z x ~! Z.

2. l/R(g) < 'r holds if and only if IAI contains a point on S(c) with c < 1, which in turn holds if and only if th~re i.~ a. ve1'le:l: ((/., b) with a.+b<1.

3.O:xyz (g) < 1 if and only if 9 . (j. (x, zt· 1. A tlerte.x of IAI lies below the line b = 1 if and only if 9 Proof. Lemma 7.15 follows directly from the definitions. Suppose that gET is such that VT(g) 1J·

~

=

L.J

'f'.

f/ (y, Z)1·.

o

rJet

biJIe:l; ~lk Y' Z

i I j IIe=r

be the leading form of g. (:r., y, z) will be called good parameters for g if bOOT'

= 1.

We now suppose that lIr(g) = r, ann (:1:, y, z) are good paraIneters for g. Let. ~ = A(g; :t, y, z) and suppose that (a, b) is a vertex of IA\. Let •

8(a,b)={kl( Define

{y }~~z

= Z1' +

•

~ k' r -J k)=(a,b)andbijk:f;O}.

r-

E

ba(r_k),b(J'_k),leXu(r k)yb(r --Ie) zk.

kES(l£.b)

We will say that {g} ~tz is solvable (or that the vertex (Q" b) is not. prepared on IA(g; x,y, z)l) if Q" b are integers and

{g }~~/Z = (z

_1]X

a

y b)'r

for some 1] t K. WA will say that. the vertex (u, b) is prepared on IA(g; x, y, z)1 if {g}~~y,z is not solvable. If {g}~tz is solvable, we can make an (a, b) preparation, which is the change of parameters Zl = Z -

If all vertices (a,b) of

IAI

''7:r. a y b.

are prepared, then we say lha,t (Y;:l:JY, z) is well

prepared.

Lemma 7.16. Suppose that'P i.fI a. pome1' /u:1"ie.s '1"iny ove1' K, gET is such tha.t vr(g) = T, ,(g) = 1, (x, y, z) are good parameters of 9, and (g; x, y,z) is well prepared. Then z = 0 is an approximate manifold of 9 = O.

7.3. r(q)

:=

115

1

Proof. lJT(g) = r implies all vertices (u, b) of ILl(g; ;z;, y, z)1 sati..qfy a+ b > l. The leading form of 9 is

'"" L-

L=

bijk X -Yi jzk ,

i+j+k=r

where bOOr = 1. T(g) = 1 implies there exists a linear form ax+by+cz such that (ax + by+ czt = D. As boo1' = 1, we must have c = 1. If a 1~(.q;x,y,z)l, which is not prepared since

{g}~~~) Thus a = 0, H prepared since

:F

0, then (1,0) is a vertex of

= (z + axt.

b =1= 0, then (0,1) is a vertex of

Thus L = zr, and z

=

°is

{g}~~;) a,1)

=

1~(gjx,y,z)l, which is not

(z + by) 7' •

approximate manifold of 9

= 0.

D

Lemma 7.17. Consider the terms in the expansion

L rl- (k) k

(7.9)

h=

A

),,-0

Xi+(k-'\)U'yi+(k->')bzf

,,\

obtained by substituting Zl = Z - T]XQyb into the monomial xiyi zk, Dejine p'fvjedion for (a, h, c) E ,N3 such that c < rand

1r(a,b,c) = (

a

.,

b

r-c r-c

(L

).

Then:

< r.

Then the exponents of monomiu.l.'!I in (7.9) with non-zero cocjficientsproject into the line segment joining (a, b) to

1. Suppose that k i -L) ( 7'_ k' r-k .

a. If (a, b) = (7' i k'

7' j

f. ) ,

then all these nwnomial.r; p1'oject to

((~,b).

b. If (a, b) :F (r~.k k)' then xiyi zt is the unique monomial in (7.9) which projects onto (t. i k' r ixJ. No monomial in (7.9) projects to (a, b). I ./

2, SUppO.Sf that r < k, and (i,j,k) #= (O,O,r). Then all exponents in (7.9) with non-ze1'O coefjident.s and Zl exponent less that r project into ((a, b) + Q;o) - {(a, b)}. 3. Suppose that (i,j, A:) = (0,0, r). Then all exponents in (7.9) with non-zero coefficients and Zl exponent less than'r project to (a, b).

7. Resolution of Surfaces in Positive Characteristic

116

Proof. If k

1= r, we have i+(k-,\)a

_

1''\

-

=

j+(k-A)b

r-X

(

+ (k -

+ (k -

A)n j

r-..\

I

we ha.ve 0 < ~_~

< A < k,

Suppose tha.t k < 'r. Since 0

't

(i ) b fr (-.L - b) , (7'-A:)

a + (7'-,\) r k - a + "-,\k~ r=J(;

< 1. Thus

A)b)

T·-..\

'

is on the line segment joining (a, b) to C,. i 11:' 7~j 11: ), and 1 follows,

°<#-

If k > rand

Thus 2 follows if r

oX

< r < k, then,~ ~ < 0,

A:.

Suppose that k = rand (

i

+- (~ __ ~)_a , j

°<

oX

< r.

(k·· ..\)b)

l

r ik -

a<

°

and

"J 11: -

b < 0.

Then i

= (

'r-A I'-A and the last case of 2 and 3 follow.

+-

(J"

'T'-A

._t_._ -+ b) ,

'r-A

0

We deduce from this lemnla that

Lemma 7.18. Su.ppose that

Zl

=

Z -

77xayb 'is an (a, b) prepa.ration. Then:

1. IA(g;x,y,Zl)1 c IA(g;x,y~z)l- {(a,b)}. 2. IJ (n' , ll) i.~ another' ver'lex of I~(!J: :1:, y, z) I, I,hen ((1,1, Ii) 'l.'l a ve1'le:l: ~f IA(g; x, y, Zl) I and {g }~:~'ZJ is obtained from {g }~:t.'z by substituting Zl /0'1" z.

Example 7.19. It is not always possible to well prepare after a finite nmnber of vertex preparations, Consider over a field of characteristic 0 (or p > r) gm

= y(y

x)(y - 2x) ... (y -

T';};)

+ (z -

x m -I- x m +1

- ...

Y

with m· > 2. The vertices of IACq;x,y,z)1 are (0,1 + ~)~ (1, ~) and (m,O), 'Ve can renlove the vertex (112,0) by the (om,O) preparation Zl = Z - :x71l • 'Thus

9 = =

y(y - x)(y - 2x) , , , (y ~ . . rx) y(y - x)(y - 2x) .. , (y -

+ (z - 11~x)r rx) + (z - x + x 3 -

,,,

r

can only be well prepared by the formal substitution 00

Zl

= z - L:)-l)ixi, i=2

Lemma 7.20. Suppose that g is reduced, VT(.q) =. r, ,(.q) = 1 and (x,y,z) are good parameters for g, Then there i.9 a formal .9cries ¢(x, y) E KUx, v]] ,fJ'lu:h /.lULl 'under' the. substitution z = Zl + r/>(x, V), (:l:, y, Z1) aT'e. good pU'T'fJ,1neters for g and (g; x, 11, Zl) is well prepared.

7.3. 7(q)

=1

117

Proof. Let 'VL be the lowest vertex of ILl(,q; x, y, z)l. Let h(VL) be the second coordinate of '1JL. Set. b = 11.( VI,). If 'VI.! is not prepared, make a vL preparat.ion Zl = Z - T/XGyb (where (a, b) = VL) to remove 'VL in ILl(g; x, y, Zl) I. Let VLI be the lowest vertex of IA(g; x, y, zl)l. If tILl is not prepared and h(17Ll) = h(VL), we Can a.gain prepare 'liLt by a. l)Lt preparation Z2 = Zl - TllXaJ yb. We can iterate this procedure to either achieve I~(g; x, y, zn) I such that the lowest vertex VI.!." is solvable or h(VD,J > h(VI.) , or we can construct an infinit.e sequence 01' 'lIL n preparat.ions (wit.h h( VL n ) -. b 1"01' all

n) Zn+l

,(in

= Zn - rlnX

b

y,

where (an, b) = VL n , such that the lowest vertex VL n of ILl(,q; x, y, zn) is not prepared and h(1}TJTI) = h(VIJ) for all n. Since an +l > an for all n, we can then rnake the formal substitution z' = z 0 ''1i:l:ai yb I to get IA(g; x, y, z')1 whose lowest vertex Vu satisfies h(v£I) > h(VL). In sUIIuuary, there exists a series (1)'(:1:) sueh t.hat. if we set z' - z yh(VL)tP'(X), and vu is the lowest vertex of IA(.q; X, y, z') I, then either vv is prepared or h(1Ju) > h(v[J 1

l::f

By iterating this procedure, we construct a series ~(x, y) such that if z = z- q,(x, y), then either the lowest vertex v,; of IA(g; x, Y, z) I is prepared, or ILl(y; :1:, y, z)1 -",. 0. This last case only or.curs if 9 = unit, z"', ami thus cannot occur since 9 is assumed to be reduced. is prepared. "~e can now apply the same IJl'Ocedur'e 1.0 VT, the highest verf.p.x of ILl(g; :1:, y, z)l, to redllr.e to 1J1t~ case where VI' and VL are both prepared. Then after a finite number of preparations we find a change of variables z' = z q,(x, y) such that (g; a:, y, Zl) is well prepared. D We can thus assume that

VL

We will also consider change of variables of the form Yl

=Y-

7JX

n

for", E K, n a positive integer, which we will call translations.

Lemma 7.21. Consider the expansion

(7.10)

(.i)

~ 'r~-A A x i+(j-A}n YlAZ k h -- L.),=0

obtained by substituting Yl = y - r]x n into the monomial x i y.1 zk. Consider the IJrojection for (a, b, c) E N3 with c < l' defined by

b 1t'(a,b, c) = ( , ). r-c r-c a

7. Resolutioll of Surfaces ill Positive Clutractel"istic

118

Suppose that k < 1'. Set (a. b) = C. i Of.' r j k). Then xiy{zk is the unique monomial in (7.10) whose exponents project onto (a, b). All othe7' monomials in (7.10) with nun-zero coefficient project to j>oint.'l below (a, b) on the Une through (a, b) with slope - ~. Proof. The slope of the line t.hrough the points (a, b) and i (

.

IS

+ (j -

A)

-X)n

r- k

'

7' -

k

j-A

1 n

i - (i + (j - A)n) Sinee r >'k

o

< 1.3. k for A < j, the lemma follows.

Definition 7.22. Suppose that gET is reduced, vr(g) = r, T(g) = 1 and (x, y, z) are good parameters for g. Let a = u:.cyz(g), /3 = (3xyz(g), "Y = "Y:.cyz (g), 6 = dxyz(g), € = €xyz(g)· Then (g; X, y, z) will be said to he very well prepared if it, is well prepared and one or t.he following conditiollil holds. 1. (')' - 6,6) :f= (a, (3), and if we make a translation Yl = y- 1]X , with 8ubseqnent well preparation Zl = Z -
(g)

= Q,

{3XYl Zl (9)

=

(3,

')'X1l1Z1 (,q) = "Y

and 6:r:Y12;1

(g) < o.

2. (, - 6,6) = (n , I3), and one of the following cases holds: a. € = o. b. (-10 and ~ is not an integer. c. f :f= 0 and n = ; is a (positive) integer and for any 1] E K, if Y1 Y ~ TJX n is a traruslation, with subsequent well preparation Zl = z - (x, y), then €XYIZ(g) = f. Further, if (c, d) is the lowest point on the line through (0, (3) with slope -{ in l~(g; x, y, Zl) I and (ell d 1 ) i8 the lowest point on thi8 line in 1.6.(g; x, Yl, ~1) I, then dl < d.

=

Lemma 7.23. Suppose that 9 E T is reduced, 1-''1'(g) =

(x, y ,z) are good ]Jammeters for g. Then Zl

= z -
tlu~re

reg) = 1 and arc formal substitution.s 'f',

Yl = Y - ,¢(x),

where 4>(x, y), '~,(x) arc series such that (g; x, Yl, zt) is very well pr·epa'f'ed. Proof. By Lemma 7.20, we may suppose tha.t (g; X, y, z) is well prepa.red. Let 0: = (}xyz (g) I {3 = (3xyz{g) , ')' = , xyz (.q), 0 = llxyz (g), E = (xyz (g) . Let ("Y - t , /.) be the highest vertex 011 I~(g;x,y, z)1 on the line a +b = "y. Let

7.3. r(q)

=1

119

K and consider the translation Yl = Y - 7JX. By Lemma 7.21, (,- t, t) and (u, /.l) are prepared verticp~<; of ILl (g; X, Yh z) I. Thus if Zt = Z - .p (x, y) is a well preparation of (g; x, Yl, z), then (, - t" f,) and (0::, (J) are vertices of IA(g; X, Vt, zl)l·

1] E

We can thus choose 11 E K so that after the translation y, = y - 7JX, and a subsequent well preparation ZI = Z - .p(~z:, V), we have nXY"'~1 - 0., /1;1:YI Z I = /J, ,3:111 Z 1 = , and 8X 1/1 Z 1 > 8 is a maximum for substitutions YI = Y - "IX, Zl = z - ep (:c, y) a.., above. rr (I - t5:1:'YlZ l ' (5:r.Y1 ZI) =1= (a., /1), then after replacing (x, y, z) with (x, YI, Zl), we have achieved condition 1 uf the lemma, so that (g; x, y, z) is very well prepared. We now assume that (g; x, y, z) is well prepared, and (0:,/.1) = ("'( - (): 6). If c = 0 or ~ is not an integer, then (g; x, y, z) is very well prepared. Suppose that n = ~ is an integer. We then choose T} E ]( such that with the traflslalion Yl - y - 'lp;'ft; and subsequent well preparation, we maximize d for points (c, d) of the line through (0, (3) with slope - f 011 the boundary of ILl(fj X, YI, zd I. By Lemma 7.21 0., {3 and '"Yare not chanp;ed. If we now have d 1= {J, we a,re very well prepared.

If this process does not end after a finite number of iteratiuns, then t.here exists an infinite sequence of changes of variables (translations followed by well preparations)

such that ni+l > ni for all i. Given n E N, there exist.s a(n) E N ~uch that a(n) > u(n - 1) for all n, and i > O"(n) implies all vertices (a, b) of I~(g; X, Vi, zi)1 below (Cit, (1) have (}. ~ n. This last condition follows since all x T. ~Z, and there are only finitely JIlany vertices must lie in the lattice ~Z 1 • point~ common to this lattice and the region 0 < a < n, 0 < b < ;3. Thus x n I ¢i(X, y-t} if i > a(n). Let i

j

j=l

k=l

L ¢j(x, y - L ''1k:J.;nk)~ W-i(:r" y) for all 'i. {Wi(X, yn is a Cauchy sequence in Wi(X, y) =

that z.;. = z Kllx, yJj. Thus z' = z - Ef\ ¢>i(X, Vi) is a. well defined series in K[[:l;, y, z]]. Set y' = V - L:~ I r/ix nj • Then IA(g; x, y', z') I has the sinp;le vertex (0, !3); and (g; x, y', z') is thus very well prepal"ed. 0

80

Definition 7.24. Suppose that gET, 9 is reduced, VT(Y) = 1', r(9) = 1, and (x, y, z) arc good parameters for g. We consider 4 types of monodial transforms T --+ Tl, where TTl is the cOlupletioll of the local ring of a monoidal transform of T, and TI has regular pararneters (:CllYt, Zl) related to the regular parameters (x, V, z) of T by one of the followinp; rules.

7. Resolution of Surfnces ill Positive Characteristic

120

Tl. Singr(g)

= V(x, Y, z), X=Xl,

Y=Xl(Yl+1/),

Z=X1 Z 1,

wit.h ''1 E K, Then gl = !r is the strict transform of 9 in Tl, and if lITi (Y1) = rand i (.ql) = 1, then (x b YJ I zd are good parameters

,

for 91-

T2. Singr(g)

= V(x, 11, z), x = Xl'Y1,

11 - 111,

Z = YIZl·

= 1L yr1

is the strict transform of 9 in T1 , and if VT1 (9\) = r alII] r(gJJ = L then (Xl, 111, Zl) are good pa.rameters for Y1. T3, Singr(q) = Vex, z), Then 91

x

= Xl;

Y

= Yl,Z = :r.lZl·

-!r

Then 91 = is t.he strict transform of 9 in T 1 , and if vT, (YJ) a.lld T(91) = 1, then (Xl, 111 I Zl) are good paramet.er'R for y"

=T

T4, Sing'r(g) = V(y, z), X = Xl,

11 = Yl,

Z

=

YIZl·

9r

Then 91 = is the strict transform of 9 in Tl' and if lITi (Y1) = r and T(gl) = 1, then (XII Yl, zI) are good paraluctcrs for 91·

Lemma 7.25. With I:hc a.ssumptions of Definition 7.24" S'upp08e that yeT is reduced, l/r(g) = r, ,eg) = 1, (:1;, y, Z) arc good parameters for 9, (:r., Y, z) lmd (:1:1, Yl, zd are related by a transfo1ination of 0'1"1£ of the above types TI-T4, and l~l(g) = r'1, 'T(91) = 1. Then there is a 1-1 COT7·e.spondenCt~

cr:

Da(g,x,y,z)~ Da(g1,Xl,Yl,Z'I)

defined by 1. (T((J.~ b)

= (a

I b - 1, b) if the transforrnc£i;oi.on 'i.'l a 'Pi with 1]

= 0,

2. a(a, b) = (u, a + b - 1) if the tran.'1fonnation is a T2, 3. o(a, I))

= (a ---

4. a-(a, b)

= (a, b -

1, b) if the transformation is u T8 1

1) if the. t7'unsformation is a

T4.

The proof 01' Lemma 7.25 is straightforward, and is left. t.o the reader,

Lemma 7.26. In each of the jO'lt1' Ca8f:,11 of the prccceding lemma, if crea, b) = (at,bl) is a. vr.rtex of IA(91iXl,YI,Zl)l, then (a, b) 'is a vertex of

IA (g; x, Y

1

z) I,

and if (g; x, Y, z) is (a, b) p1-e.pa7·tid I,hen (Y1; Xl, Yl, zd is (a 1, bd prepa1'ed. Tn pU1'lic'lJ.la1', (Y1 ; Xl, Yl, Zl) is well prepared if (y; x, y, z) is well prepared.

7.3. -r(q) = 1

121

Proof. In all cases, u (when extended to JR2) take~ line segments to line segments and interior points of I~(gj x, y, z)1 to interior points of u(I~(g; x, y,

z)1).

Thus the bounda.ry of (1(I~(gj X, y, z)l) is the union of the irnage by (T 01" the line segments on the boundcu'Y 01" (1(I~(g; x, y, z)l). If (all bt) is a vertex of (T(I~(gj :c,y, z)/), it must then necessarily be u(u, b) with (a., b) a vertex of u(I~(o9; x, y, z)l).

To see that

(.Ql;XI,YI,Zl)

is (al,bI) preparF.o if (gjx\y,z) is (a,b) pre-

pared, we observe that 1

{I

}a,b

xr y ·7:,lI,Z

if we perform a T3 transformation or a Tl transformation with 7J = 0, if we perform a T2 or T4 transfOl·mation. [J

Lemma 7.27. /:htppo.'1e that assumptions are as in Definition 7.24 and (x, y, z·) and (Xl, 111, '!l) are related by n T.9 transformation. Suppose that lIT! (yd = rand 7(g}) = 1. If (09; x, Y, z) is ver'Y 'Well prepared, then (gl; XI, YI , ZI) is ve'ry well p7't:pared. Marcover,

/JXl1/l Zl (91)

=

,l3xyz Cq),

8XIY1Z1

(gd = 6xyz (Y):

f:r.U/lZl

(91)

= Ca;yz(g)

and UX1Y1Zl

(91)

= O'xyz(Y) -

1,

1'xl1J1Z1

(gt)

= I'xyz(g)

- 1.

We further' have Singr (91) C V(Xl' zt}.

Proof. Well pl'epara,tion is preserved by Lemma 7.26. Suppose (y; ~l:, y, z) is very well prepared and (U X1JZI

(3xyz)

I- ('/'Xy: -

§XYZl

t5 xyz ).

Let I' = '/'xyz(g) , 1'1 = 'Yxyz{gI) - ( - 1. 'l'he terms in 9 which contribute to t.he line Fi(() arc

L

(7.11)

i

+ j + 'Yk - 'rr k
They are transformed to

(7.12)

"L.(i + k - r) + ,j + An k

k
b.. ".i+k-r'/jz k k 'l..1 ·'" 1 III l '

= r'Yl

122

7. Resolution of Surfaces in Positive ClIaracteristic

whieh are the terms in gl which contribute to the line S(~Y1). We have

.

Oxyz(g)

= «5X l,Yl,Zl (gl)

= min{ r

~k

Ibijk

1= 0 in (7.11)}.

!fa translation y =y +7]x transforms (7.11) into

"'.J .

'. ' k (''ijk X 'iF Z ,

L

(7.13) -i

+ j + "'fA: = k
then the translation Yl = Y1

'1',

+ 1]X transforms (7.12) into

L

(7.14)

(i

+ k .- r) + j + 1'1 k = r1'l k:<1'

Since (7.13) and (7.14) are the terms which contribute to o:cyz(g) and C5X1YIZ1 (91), we have 8xyz (g) = c)XWIZl (91) and (9; x, y, z) well prepared implies (gl; Xl, Y1, zd well prepared. \Ve can make a similiar argument if

(a XYZII3XYz)

=

(,xyr - OXYXI OXYz).

The statement Singr(gI) c Vex}, Zl) follows since Singr(g) = V(:L~, y, Z), so that Singr (91) C V(xt}, the exceptional divisor of T3, and Sinp;r(gt} C V(Zl) since z = 0 i" an approximate manifold of 9 = [) (Lemmas 7.16 and 7.5), D

Lemma 7.28. Suppose that assumptions are as in Definition 7.241 (x, y, z) and (Xl, Yl, zI) are rdated by a T lt1'an,,'Iformation with 7] = 0, (9; :2:, Y, z) i8 very well prepared and l/Tl (gl) = r, r(gd = 1. Let (gl; Xl, y', z') be a very well preparation of (91; Xl, YI, zt). Let

a = axyz(g), a1

f3 = f3xyz(.q),

= aXlylz,(gt), 01

1 = Ixyz(g),

/31

0 = OXyzCg) ,

= /3xlY'Z,(gt},

= Oxw' z, (gt),

(1

€

= €xyz(g) ,

11 = l'X1Y'Z,(g1),

= (;:r.1Y' z, (g1 )•

7·lL",en 1

(/11, 81, -

1

I

~1

t'Y.1) < (/1, c),-, a) ~

in the lexico.qraphical order, with the convention that if € If /31 = /3, 01 = 8 and € # 0, then = ~ - 1.

t

We further have Sing'I'(91)

c

V(Xl, z').

=

0, then 1t =

00.

7.3. r(q) = 1

123

Proof. Let a : ~ (g; X, Y t z)

~ (gl ; Xl, Yl, Z1) be the 1-1 correspondence of Lemma 7.25 1, which is defined by a(u, b) = (u+ b-l, b). a transforms liues --t

of slope m #- -1 to lines of slope vertical lines.

m~ 1 ,

and transforms lines of slope-l to

First suppose that - f < -1. Then (Ct:, (:J) i- ("Y - d, d), and we are in case 1 of Definition 7.22. Then the line segment through (a,j3) and (" - 0,0) (which ha..l;J slope < -1) is transformed to a seglnent with positive slope, or a vertical line. Thus /3x 1 ,'Yl ,ZI = 0 < /3. Since (g; Xl, Yl ,zd is well prepared, after very well preparation we have /31 < /3. Suppose that -1 < - f < -~. We have «(}I,/l) = (, - 8,8). Then the line segment through (a, {3) and (c, d) (with the notation of Definition 7.22) is transformed to the line segment through (0: + 13 - 1, /3) and (c + d - 1, d), which has slope -f

-(XI111 Z 1

=

-t+1

<

-l.

Thus (O'XI Y1 Z1

'#£1 Yl Zl )

= (0' + f3

- 1, (3)

and (l'Xl'Yl%l

(gd -

OX1'Yl%1

(gd, 6X1Y1 %1 (gd)

=1=

(aXIYIZI

(gl), (:JX1Y1 Z l (gd)·

(gI) - t, t) be the highest point on the line S( rX IYl%l (ql)) in 1~(!Jl; Xl, Y1, Zl) I· By Lemma 7.26, (91 j :];1, Yl, Zl) is well prepared. Thus (by

Let

(,XIU1 Z1

Lemma 7.21)

very well preparation does not affect the vertices (aXIYIZl (gl), f1xIYIZl (gl)) and (')'XI111Z1 (g1) - t, t) of 1~(91; Xl, Yl, Zl) I which are below the horizoutalliJle of points whose second coordinate is d. Thu..<; /31 = /3 and 61 < t < d < O. Suppose that ._~ < - f < O. hl this case we have (a, /3) = (')'-6,6). Then the line segment through (a, (3) and (c, d) (with the notation of Definition 7.22) is transformed to the line segment through (Q+/3-1, /3) and (c+d-l, d) whidl has slope -€

-€XIYIZl

so that -1 (O'X1

<-(XI1/1 ZI

YI Z1

,

= _( + 1 '

< O. Thus

{lx IY1 Zl )

=

("YXl YIZI

-

6X1 Yl Z1 , ()Xl 'Yl Zl)

=

(a + /1 - 1,~),

and (CIt d) = (c + d- I, d) is the lowest point on the line through the point (o.XIYIZ1,{1xIYIZl) wit.h slope - t x1 'YIZl which is on 1~(gl;XltYl,ZI)I. By Lemma 7.26, (gl; Xl, Yl, zt) is well prepared. We will nuw show that. (gl; Xl, Yt, Zl) is very well prepared. If E~lt!l"~l 1 ¢ N then this is immediate, so suppose that 1 llI.T n= - - E l'~. 1

7. Resolution of Surfaces in Positive Characteristic

124

We neees:sarily have n > 1, and ~ = n + 1. If we can remove the vertex (cl,d) from 1~(gl;XI,Yl,ZI)1 by a translation Yl = Yl - 'fp:i, then the translation y = Y - 7P:'Tt+ 1 will remove the vertex (c, d) from I~ (g; X, y, z) I, a euntradictioll to the assumption that (g; x, y, z) is very well prepared. Thus (gl; Xl, Yl, Zl) is very well prepared, with 1..=1_l. E1

/31

=

X},

YI, Zl) is very well

/1, 61 = 6 and

f

The final ca..,e is when prepared,

f

= O.

Then f:J = 0, (gl;

(O:bJh) = (')'1- (h,lh) = (0 + {1-1,(J) O. Thus /31 = {3, (h = d and 1 1 = ~ = 00. Also,

and €1 = Singr(g) = V(x, y, z) (by assumption), so

€

=

0'1

< u, since

0 implies {3 < 1.

Dy Lemmas 7.4 and 7.16, Singr(Yl) C V(:1:1; Z1)' HSingr(gl) = V(Xl' Zl), then all veltic~s of I~(gl; Xl, Yl, zl)1 have the first cordinate > 1, and thus Sing.,.(.Ql) = V(Xl, z'). 0

Lelnma 7.29. Suppose that assumptions are as in Definition 7.24, (x, Y, z) and (Xl, Yl, ZI) are related by a T2 or a T4 transformation, and (g; :1:, y, z) is well prepared. Suppose that lITl (91) ~ 'f' (m.d r(gt) = 1. We have that (gli:1.:l,Yl,Z1) i.':J 'Well pr'epared and Sing.,.(gt} C V(Yl,ZI). Let (gt;Xl,Y',Z') be a very well preparation. Then j-JX1 ,y, ,%1 =

f3Xl

,y' ,z' (gd < (3xyz (g).

Proof. First suppose that T ---7 Tl is a. T2 traIl~formation. The correspondenee (J : ~(g; X, y, z) --+ ~(gl; Xl, Yl, zt} of Lemma 7.25 2. is defined by O'(a, b) -' (a, a + b-1). 0' takes lines of slope 7n to lines of slope Tn + 1. Thu:s (U X1 Y1 %,

(gl), (3xIYIZl (91)) = (axyz(g), axyz(g)

+ (3xyz (g) -

Since Singr(g) = V(x,y,z) by a,l;jsurnptiou, we have O!xyz(g) ;3X1YIZ1

1).

< 1, and thus

(gt} < I3Xyz(g).

(g1; Xl, Yl, ZI) is well prepared by Lemma 7.26. Therefore, the vertex (aX11l1Z1

(g1), 11:1'.IY1z1 (gl))

ilS not affected by very well preparation. We thus have I3xIU'ZI(g,) < f3:r.yz(g). Now suppose that T' ---7 1 1 is a T4 t.ransforInatioIl. The 1-1 corresponclenr.e rr : ~(g;:,{:,y,z) --+ ~(gl;Xl,YI,ZI) of Lemma 7.25 4 is defined by O'(a,b) = (a,b-I). Thus /3xIYIZ1(gl) < {3xlI z (g). ALlID, (YliXl,Y1,ZI) is well prepared by Lenuna 7.26, and the vertex (OXIY1 Z1(gt},!3X1Y1Z1(gt}) is not affec:ted by very well preparation. Thus ;3Xly'Z'(Yl) = (3X1YIZl < f3xyz(g). D

Lemma 7.30. Suppose that IJ inrt a lwim.e, s is a non-negative integer and ro i:~ a positive integer such that p f ro. Let r = raps. Then:

7.3. T(q) = 1

1. 2.

125

(D = 0 mod p, if pS f A, 0 < A< r is an integer. ex;.'!) =(~) mod 1 if 0 ~ A "5 "'0 is a.n integer'. J,

P roo f. Com paTe the expansiuns over Zp Qf (x + y)T = (x P8

+ yp

tl

)To .

0

Theorem 7.31. Suppose that assumptions are as in Definition 7.24, (x, y, z) and (:r.] , Yl, Zl) are rela.ted by a. T 1 l1unsf01"mation with r/ # 0, and (g; x, y, z) is very well prepared. Suppose that lI'l'l (.Ql) = r, r(91) = 1 and I3xyz (g) > O. Then there exist good parameters (x, yi , z~) in 11 .'1uch that (91 ;:z: 1, 'U~ 1 Z~) i.'3 1Jen/'well prt~pared, with /1X1lY i,zi (gl) < (:Jx,y,z(g) and Singr(gI) C V(XI, zD. Proof. Let 0' = UX'IIZ (g), f:J = f:Jxyz (g), 'Y = 'Yxyz (g), 8 = Oxyz (.9). Lemma 7.16 implies z = 0 is an approximate manifold of g = O. Thus

a+(J>l.

(7.15)

Apply the translation y' = y - 7]X and well prepare hy some subsLitutiulI z' = Z - (:z:,y'). This does not ehange (n,f3) or "y. Set 8' = 8x~y/,z'(.9). First assume that 0' < 13. We have regular parameters (Xl, Yl: 21) in Ti such that x = Xl, y' = XIYl, z' = :l:lZl by Lemma 7.16, su we have a 1-1 currespondence (by Lemma 7.25 1)

a : ~(g; x, y'; Zl)

--+

~(91; Xl, YI, Z1)

defined by a(a, b) = (a+b-1, b). (gl;Xll Yl, Zl) is well prepared (by Lemma 7.26). We are essentially in case 1 (-(; <>~ 1) of the proof of Lemma 7.28 (t.his case only requires well preparation), su that (0:'1,131) = a(1' - 0',8') and 131 = 8' < (3. After very well preparation, we thus have regular parameters Xl, yi , zi in T1 such that (91 j x1, y~ , z~) is very well prepared and PX!,y~,Z~ (91) < /-Ixyz(Y)· We have thus reduc:ed t.he prouf tu shuwing that 8' < ;3. If (a, (3) ('Y - 8,0), we have 8' < 8 < /3 since (g; X, '11, z) is very well prepared. Let g

= L aijkxiyi zk.

Suppose that ((Y.,

In = (, .

.

W =

Hi,j, k)

E N

3

1k < r

F =

and

L

8, 8). Set

(1'~ k' ',. ~ k)

= (o,{3)} 1

ai.1kxiyJ zk .

(i,j,z)EW

By assumption, zr

(7.16) F(x, y, z)

+F

is not solvable. J\1oreover,

= F(x, y' +'IX, z) =

L

t a;jk'l~ G) xi+A(y')j-~zk,

U,j,k)EW).=O

=1=

126

7. Resolution of Surfa.ces in Positive Oharacteristic

By Lemma 7.21, the terms in the expan..,ion of g(x, y' to (')',0) in IA(q; x, y', z) I, where l' = Q + {:J, are ""'"' ~

Fb,O) _

. .x i+'J z k (lijkrr

+ 1]X, z)

contributing

# n.

(i,i,k)EW

If (g; X, y', z) is (/,,0) prepared, then 6' = 0

< 13.

Supp0l:>e that

{,q} ~~/,z = zr + Fh,o) is solvable. Then lEN, and there exists '¢J E K such that

(7.17) 80

that, with w =

L If

"1 E: K, for 0 < k < r, we have:

(r:AJ #- 0

(in K), then i =

(7.18)

aijk =

''1'

(

k),j = /3(0,. - A;)

0:(7' -

r _ k

)

"-k

W

.

2. If·;, = a(r- k),j = j1(r - k) E N and aijk

E Nand

(r:k) =

0 (in !(), then

= O.

Thus (by Lenlma 7.30)

ai,ik

= 0 if pS t k.

Suppose that K has chara.cterist.ic O. By R.emark 7.:-12, we obtain a contradiction to our assumption that {g} ~~I~ is solvable. Thus 0 = 6' < Ii if K has characterist ic O. Now we consider the case where K has characteristic p > 0, and 'r = ll''1'o withpfro, TO > 1. By (7.18) and Lemma 7.30, wehavei.=. Q:pS,j = {3ps EN and ai,j,(ru-1)pS # O. We have an expression (Jps = ept, where p f e. Suppose that t > s. Then {3 E N, which implies Q = I - {3 E N, 1:10 that

(z

+wxO:yfJt = zr + P,

a contradiction, since zr + F is by asl:lUmptioIl not solvable. Thus t < s. Suppose that c = 1. Then {3 = pt-s < 1 and Q < 1 (since we must have Sing,. (g) = V (x, y, z)) imply l' = (}; + IJ = 1, a contradiction to (7.15). Thus c > 1. Also, ZT

+F

+ wPIi X Olpli y{3pll )ro (zPS + w ps x ops (y' + '17 X)f31>!l) ro (zps + WP8XClp8 [(y,)pt + 77pt x pt ]e) TO •

(Zpll

_

7.3. T(q) = 1

127

Now make the (,,0) preparation z = z'- 7](lwx7 (from (7.17)) so that (g; x, y',l) is (ry, 0) pff~parf',d. Let G = zr + F. Then

G

((Z')PIl

+ ewP·97]pt(e-l)(y'JP"xaP"+p"(e.-l)

o

I (y')2 ptn ))r

(z,)p"ro + ro [ew P"7]p t(e-l) (y,)pl xap"+pl(e-l)

+ (yl) 2pt n]

(z,)p·q(ro-l)

+A2(:r., y')(z,)p"(ro-2) + ... + Aro(:r:, V') for some polYllOlnials n(:1:, V'), A2, ... ,A"01 where (yl)ipl I Ai for all i. Since all eontributions to S(1') n I~(gj x, y', z')1 must come from this polynomial (recall that we are assuming (a,.8) = ('Y 8,8)), the term of lowest seeond coordinate on 8(,) n ILl(g; ;1:, V', zl)1 is

(a.,. b) - (

a pS

+ li(F.. pS

1) ,pt ,) , pS

which is not in N2 since t < s, and is not (a,p) since e > 1. Thus (g; X, is not (a, b) solvable, and

y', Zl)

o Remark 7.32. In Theorem 7.31 suppose that K has characteristic 0. From (7.18) we see that if {q};~,z' is solvable, then for < k < r 1 there exist i,j E N with i = a(r - k), j = (3(r - k) so that el, j3 E N. Comparing (7.18) with the binomial expansion

°

we se<:: that zl'+F is solvable, a contradiction. Thus (1',0) is not solvable on I~(g; :1:, V', z)1 irK IJal:! characteril:itic 0, and after well preparation, (,,0) is a vertex of ICq; X, y', z') I. Example 7.33. With the notation of Theorem 7.31, it is possible t.o have (,,0) solvable in [d(g;x,y',z)1 if K has positive characteristic, as is shown by the following simple eXaIllple. Suppose that the characteristic p of K is greater than 5. 2 :l Let 9 = zP + ;l;'YP 1 + xV -ly2. ~

The origin is isolated in Singp (g = 0), and (g; x, y, z) is very well prepared. The vertices of Id(gj X, y, z)1 are (a, {3) = C"Y - 6, 6)= (~,p - ~) and

7. Resolution of Surfaces in Positive Characteristic

128

(p - ;, ;). Set y = y' +

fIX

wit.h 01="1 E K. Then

+ x(y' + "I X )p'l-1 + x P2 l(y' + 1]x)2 2 zP + (2:r o 1 (P:l;1)1JP2_1-ixP2_i(y')i) +71 2X P2 +1 + 2rlY'xP2 + :r,p 2_ 1 (y')2

9

zP

I~(g; X, y', z)1

The vertices of

are (a,.B) = (;: P -

However, ('"Y, 0) is solvable. Set z! g

= (z')P+

= Z + '1

P

-

112 (

it follows that

(a,.B) = (~,p-

+7]2XP2+1+2''1Y'~l:p2+x P2 - 1(y')2.

't,

Since 1

1) =

-1

(P, 0).

Px'P. I'hcn

i

'i=l

and (1',0) -

1

2:-1 (21J .- 1)".,p2-1- x P'l-i(y')i

p2

~)

mod P,

01 = 8xyIZl(g) = ~, a.nd the vertices of IA(g; X, y', z')/ are

;J, ("'fI -01,lh) -

(p- ~,~) and (p+ ;,0).

Theorem 1.34. For all n E N I the're a.re in E R n such that In is a generator oj' TSn.I'ltl. Rn., and good parameters (x n Yn Z'TL) 1m' !n in R n such that if I

f3n = !3xnY'f)Zn (!".) , Un -=

(}XnlJ,.,z.~I(ln),

I

f. n

=

(X"Y'IZn (f71.)'

671. = 6XnYnzn(in),

2.-,

u(n) = (!3n,6n , On), , €n then a(n + 1) < u(n) 10'1' a.ll n in the lexicographical order in~N x ~IN x (Q+ U {oo}) x :IN. 11#11.+1 = J~n, 671.+1 = On and F~' 1= Xl, then

1 1 --=--1. en-II

fn

AR an iInmcdiate corollary, we find a. contradiction to the assumption that (7.8) has inllnit.elength. We have now verified Theorem 7.9 for the final case r( q) = 1.

Proof. We inductively construct regular parameters (x n ? Yn, zn) in R It , 8 generator in of ISn,q.IlRn such that (x n Yn, z-n) are good parameters for In, Sing'I'(ln) C V(x n , zn) or Singr(ln) C V(Yn, zn) and (in; X n , YTII zn) if; well prepared. If Sing,. (!n) C V(x n , zn), we will further have that (In; Xn, Yn, zn) ?

is very well prepared. Let .f be a generator ol''Ls,!JRo. We first choose (possibly fornlaJ) regular parameters (x, y, z) in R which are good for f = 0, such that IA(I; x, y, z)1

7.3. T(q) = 1

129

is very well prepared. By Lemma 7.16, Z = 0 is an apprnxhnate nlanifold for f = O. Let Q' = i'i:r;,y,zU'), f~ = 11.r,,1J,z(f)· Hy assumption, q is isolated in Singt.(f).

Suppose that Ji and regular parameters (Xi, Yi, Zi) in fined for i < n as specified above.

R i have been de-

= V (Y1H zn), then Rn -+ Rn+l mut\t be a T4 Lrflllsfonnation:

If Singr(fn)

r, :,'11.

=,Xln+1'

Yn = .'#n+1, 1/

-- 'y'n+1 z' ....~'n -''Il.+I·

Thus (In I 1; :l:"1+ 1 ~ Y;l+1' z;~+ 1) is prepared by Lemma 7.29. Fu rt her, Singr(fn+t)

C

V(Y~+l' z~ 11)'

V(:1;71+1'Y~,+1'Z~+1)' make a change of variables to very well prepare. Otherwise, set YTL I 1 -- Y'~},+1, Zn+ 1 = Z~+ l ' We have {3n+ 1 < ;J.n by Lernrna 7.2H.

If Singr(fnl d

=

If Singr(!n) = V(x 1tl zn), then Rn -----.. R n+l must be a T3 tram;formation Xn = Xn

III

Yn

= Yn.+l,

Zn = Xn+lZn+l,

and (,fn;xn,Yn,zn) is very well prepared. Thus (fn+l;X n+1,Yn+l,Zn+l) is also very well prepared. Nloreover,

(,8n +1,On+],

1

£n+1

1

,un+d < (I~nl 8n , - , O:n) En

by Lemma 7.27. Further, Singr(!n+d C V(X n +l1 zn+d. Now suppose that Singr(,fn) = V(x n , Yn, zn). Then (/; Xn, Yn, 2'11,) is very well prepared, aJld Rn +l mllst be a Tl or T2 transformation of Rn . If R.n+ 1 is a T2 transformation of R,~ 1

I

Zn

= Yn+lZn+l1

then (fn+l;Xn+l'Y~+1,Z~11) is well prepR.l'ect with 1311,+1 < 7.29. Further, Singr(fn+l) C V(Y~+l' Z~+ 1)' If

13n

by LeII1Iua,

Singr(fn+l) = V(X 11,+1, Y:t I 17Z~'+1 ), then make a further change of variables to very well prepare. Otherwise, set ,

I

Yn+1 = Yn +l' Zn+l = Zntl'

I1'RlI +1 is a Tl transformation of :1:11, =

:1:11.+1,

Rn,

Yn = Xn+l (Y~4'1 + 11),

Zn

=

X 1l +1 Z~''+l'

then Singr (!n+l) C V(x n +ll Z~+l) by Lemma 7.5.

11 i 0 in the Tl transformation relating Rn and R n + I, we can change variables to Yn+l, Zn+l to very well prepare, with ,6n+1 < ,8n. and Singr(!n+d c V (:r11.+h Zn+l) by Theorem 7.31. If

f3n :f:. 0 and

7. Resolution of Surfaces in Positive Characteristic .

130

,

If TJ = 0 in the Tl transformation of R n , and (Xn+b Yn+l, Zn+l) are regular parameters of R'll+ 1 obtained from (;7:11.+ 1, V:7+ l' Z:7.+ 1) by very well preparation, then (In I 1; ~Z:n 11, Yn+ 1, Zn+l) is very well prepared,

(,8n +1 ,8n -l- 1 , 1 ,On+l) < €n+l

(;3n,8n,~,Q:n), . En

and Singr(fntd c V(:l:'fd 1, Z'fL I d by LeIlllna 7.28. 0, we can make a translation y~ = Yn + 'l1Xn, and have that (g~;r,n, y~, 211.) is very well prepared, and /In , '"'tn, ~'n' tn, (}.n are unchanged. Thul:! we are in the case of'TJ = O. 0

If (3n

=

7.4. Relnarks and further discussion The first proof of resolution of singularities of surfaces in positive character-

istic was given by Abhyankar ([1], [3]). The proof used valuation-theoretic methods to prove local uniformization (defined in Sedinn 2.5).

The nlOst general proof of resolution of surfaces, which includes excellent surfaces, was given by Lipman [63]. This lucid article gives a self-contained geometric proof. This proof of resolution of surfaces which are hypersurfaces over an algebraically closed field (Theorem 7.1) extends to give a proof of resolution of excellp,nt surfaces. Some general rernarks about the proof are given in the final 3 pages of Hironaka's notes l53J. The first part of our proof of Theorem 7.1 generalizes to the case of an excellent surface S, and we reduce to proving a generali~atirm of Theorem 7.9, when V = spec(R), where R is a regular local ring with maximal ideal m and S = spec(R/ I) is a two-dimensional equidimcnsionallocal ring with Singr(R/ I) = V(m).

Tllere are two main new points which must be considered. The first point is to find a generalization of the polygon IA(f; x, y, z)1 to the case where OS,q r-v R/I is as above. This is accomplished by the general theory of characteristic polyhedra [54]. The second point is to extend the theory to the case of 8. non-dosed residue field. This is aecomplished in [23].

We will now restrict to the case when our surface S is a hypersurface over an arbitrary field K. The proof of Theorem 7.9 in the case whp,n T(q) = 3 is t.he same, and if r( q) = 2, the proof is really the same, since we must then have equality of residue fields !«(qi) = K(qi+l) for all i in the sequence (7.:i), under the asslunption that r(q'i) = 2, vql. (Si.) = 'f' for all 'i. This is the analogue of our proof of resolution of curves embedded in a non-singular surface, over an arbitrary field. In the ease when r(q) =- 1, smile extra care is required. It is possible to have non-trivial extensions of residue fields k(qiJ -+ k(qi+l) in the sequence

7.4. Rem8,rks and fur·tber discussion

131

(7.8). However, if k(q·i.+l) =1= k(qi), we lllURt ha.ve j3'i+l < #i, (This is proven in the theorem on page 26 of [23]). Thus ({3i,bi, ~, Qi) drops after every transformation in (7.8), and we Ree that we UlU:;t eventually have 1/ drup ur T increa:;e. The essential difficulty in resolution in positive characteristic, as opposed to characteristic zero, is the lack of existence of a hYI)er'surl'ace of maxiInal contact (see Exercise 7.35 at the end of this section). Abhyankar is able to finesse this difficulty to prove resolution for 3dimensional varieties X, over an algebraieally dosed field K uf characteri:;tic p > 5 (13.1, 14]). We reduce to provin~ local uniformization by a patching argument (9.1.7, [4]). The starting point of the proof of local uniformizat.ion is 1.0 use a projection argument (12.4.3, 12.4.4 [4]) tu reduce the problem to the case of a point on a normal variety of multiplicity < 3! = 6. A sinlple exposition of this projection method is given on page 200 of [62]. We will :;ketch the proof of resolution, in the case of a hypersurface singularity of multiplicity < 6. We then have that a local equation of this singularity is

(7.19)

'Ut

+ ll.7'-1 (:1:, y, z )'W

T

-

1

+ ... + llO(X, y, z) =

O.

Since r S 6 < char(K), it follows that ~ E ](, and we can perfornl a Tschirnhausen transformation to reduce to the case

w + ar-:l (x Y, Z )w 'r'

I

0

1

-2

+ . . . + au (x, Y, Z)= 0 .

Now w = 0 is a hypersurfacc of maxinlal contact, and we have a good theorem for embedded resolution of 2-dirnensional hypersurface:;, l:iU we reduce to the case where each ai is a monomial in x, y,Z times a unit. Now we have reduced to a combinatorial problenl, which can be solved in a characteril'ltic free way. A resolution in the case where (7.19) has degree p = char K is found by Cossart [24]. The proof is ext.raordinarily di1ficult. Suppose that X is a variety over a field K. In [36], de Jon~ has shown that there is a dominant proper morphism of K-varieties X' .-. X such that dim X' = dim X and X' is non-singular. II" K is perfect, then the finite extension of function fields K(X) --+ K(X') is separable. This is weaker than a resolution of singularities, since the map is in general not birational (the extensiuIl uf fUIlction fields K(X') --+ j{(X) il:i finite). This proof relies on sophisticated methods in the theory of moduli spaces of curves.

Exercise 7.35. 1. (Narasimhan [69]) Let ]{ be an algebraically closed field of char-

acterisitc 2, and let X be the 3-fold in At- with equation

f = w2 + xy3 + yz3 + zx 7 =

O.

7. Resolution of STllfa.ces in Positive Characteristic

a. Show that the maximal nulltiplicity of a singular point UII X is 2, and the locus of singular points onX is the monomial curve C with local equations

y'j + zx 6 = 0,

xy2

+ z:l =

0,

0, 'UJ2 + zx7 = 0, which has the parameterization t -+ (t 7 , t 19, t 15, t 3'2). Thus C has embedding dimension 4 at the origin, so there cannot exist a hypersurface (or formal hypersurface) of maximal contact for X at the origin. b. R.esolve the singularities of X. yz2

+ :1/ =

2. (Hauser [50]) Let K he an algebraically closed field of characteristic with equation f = x 2 + y4 z + 2, and let S be the 8urfaA~e in y2 z 4 + z7 = O. a. Show that the maximallllllltipliCity of points on Sis 2, and the. singular locus of S is defined by ,the singular curve y2 + z3 = 0,

Ak

x

+ yz2 =

O.

b. Suppose that X is a hypersurface on a non-singular variety W, and p E X is a point in the lucus of maximal multiplicity r of X. A hypersurface H through p is said to have permanent contact with ..¥ if under any sequence of blow-ups 1T" : WI ~ W of W, with non-singular centers, eontained in the locus uf point.s of multiplicity r on the strict transform of )(, the strict transform of H contains all points of the intersection of the strict transform of X with multiplicity r which are in the fiber IT··! (p), Show that there does not exist a non-singular hypersurface of permanent. contact at the origin for the above surface S. Conclude that a hypersurfa.ce of maximal contact dues not exist for 8. c. Resolve the singularities of S.

---Chapter 8 Local U niformization and Resolution of Surfaces

In this chapter we present a proof of resolution of surface singularities through local uniformization of valuations. Our presentation is a nlOdernization of Zariski's original proof ill [86] and [88]. An introduction to this approach wa..~ given earlier in Section 2.5

8.1. Classification of valuations in function fields of dimension 2 Let L be an algebraic function field of dimension two over an a.lgebraically closed field K of characteristic zero. That is, L is a finite extension of a rational function field in two variables over !(. A valuation of the function

field L is a valuation lJ of L which is trivial on K. v is a hOlllomorphisIll v: L* -. r frOIlI the multiplicative group of L onto an ordered abelian group

r

such that

1. v(ab) = ll(a) 2. v(a + b)

+ ll(b)

for a, bEL *',

> uun{v(a), v(b)}

for a, bE L*,

3. lI(e) = 0 for 0 1: e E K. We formally extend II to L by setting 11(0) = 00. We will this section t.hat v is non-trivial, that is, r =1= O.

a.QSllme

throughout

Some basic references to the valuation theory of algebraic function fields are [92] and [3}.

133

8. Local

134

Uniformi~atic.JIl

ana Resolution of Surfaces

A basic invariant of a valuation i8 its rank. A subgroup r' of r is said to be isolated if it has the property that if 0 < ~ E r' and f3 E r is such that o < (3 < ll:', then fl E r'. The isolated sllbroups of f form a single chain of subgroups. The length of this chain is the rank of the valuat.ion. Since L is a two-diluensional algebraic function field, we have that rank(v) < 2 (cf. the corollary and the note to the lemrna of Section 10, Chapter VI of [92]). If f is of rank 1 then r is embeddable as a subgroup of lR (cf. page 45 of Section 10, Chapter VI [92]). If r is of rank 2, then there is a non-trivial isolated subgroup r 1 C r l:mch that f 1 and r Ir 1 are embeddable a.~ subgroups of JR. If these subgroups of R are discrete, then 11 is called a discrete V'dluatioll. JTI part.icular, a discrete valuation (in an algebraic function field of dimension two) can be of rank 1 or of rank 2. The ordered chain of prime ideals p of V corresponds to the isolated subgroups r' of r by

r - r'} U {OJ

P = {f E V Iv(f) (

(cf. Theorem 15, Section 10, Chapter VI [92]). In our analysis, we will find a rational function field of two variables L' over which L is finite, and consider the restricted valuation v' = v I L'. We will consider L' = K(x, y), where x, y E L are algebraically independent eleluents such that v(x) and ll(Y) arc non-negative. Let [L' : L] = r. Suppose that W E L~ and suppose tha.t

(8.1)

F(x,y,w) = Au(x,y)w'" + ... + A.,.(:l:,y) =

°

is the irreducible relation satisfied by w, with Ai E ]
r'

Associated to the valuation

v= V

h~

11

r

is the valuation ring

{f ELI v(f) > O}.

a unique maxima.! ideal

mv

=

{f E V

1

,

/(f) > O}.

The residue field [((V) = Vlmv of V contains]( and has transcendence degTee r'" < ". over K. v is called an '1'* -dimeIL'3ional valuation. In our case (v non-trivial) we have r· = 0 or 1.

If R is local ring cont.ainedin L, we will say that v dominates R if ReV and mv n R is the maximal ideal of R. If S is a 8ubring of V 1 we will say that the center of v on S is the prime ideal mv n S. We will say that R is all algebra.ic local ring of L if R is essentially of finite type over K (R is a localization of a finite type ]<-algebra) and R has quotient Held fJ.

8.1. Clt:tssific.:ation of valuations in function fields of dimension 2

135

We again consider our algebraic extension L' = K(x, y) ~ L. The valuation ring of v'is V' = V n L, with maximal ideal mVl = mv n L. The residue field K(V / ) i~ th11s a subfield of K(V). Suppose that w* E K(V), and let. w be a lift of w* to V. We have an irreducible eqnation or algebraic dependence (8.1) of w over K[x, V]. Let AlL be such that

v(A n.)

= min{v(AI), ... , v(A7,)}.

We may assume that h #- T, becam;p. if v(A 1·) < 11(Ad for i < r, then we would have v(A r ) < v(Aiw r - i ) for 1 < i < r - 1, since v(w) > 0, so that. l/(F) = v(A r ) < 00, which is impossiblp.. Thus v(~) > 0 for 0 < i ~ r, so that ~ E V' for all i. Let have a relation

aMw*r + ... + ail

l(W*Y'

at

be the corresponding residues in j«(V ' ). We

hll

+ (w*(-h + ah+l (-w*)r-h-I + ... + (1,:' =

O.

Since.,. - h > 0, w* is alg;ebraic over K(V'). Thus v and v' - v I L ' have the same dimension. We will 1I0W give a classification of the various types of valuations whieh occur on L. 8.1.1. One-dimensional valuations. Let x E L be the lift of an element of K(V) which is transcendental over ](. Suppose that every y E. L which is t.ranscendental over K(x) satisfies v(y) = O. Then v must. be trivial when restricted to K(x, y) for any y which iR LraIlseendental over K(x). Since L is finite over K (x, y), we would have that v has rank 0 on L, so that v is t.rivial 011 L, a contradiction. We may thus choose 'Y E L such that J:~, y are algebraically independent over K, v(:n) = 0, l/(Y) > 0 and the residue of x in K (V) is transcelldental over If. Let L ' - K (x, y), and let v' = v I L' with valuation ring V' - V n L ' . By construct.ion, if '{nV' is the maximal ideal of V', there exists c E K such that TnVl nK[x,yJ c (x-e,y). Suppose (if POfifiible) t,haL 11 -= rTLVI n I<[x, yJ = (x - c, y). Then the residue of :z; in K(V) is c, a contradiction. Thus p is a height one prime in K[x, y] and p =- (y). It follows that t.he valua.tion ring V'is R p (d. Theorem 9, Section 5, Chapt.er VI [921) and the value group of v'is Z. The valuation ring V is a localization of the integral closure of the local Dedekind domain V' in L. Thus V is itself a local Dedekind dornain which is a localization of a ring of finite type over K. In particular, V is itself a regular algehraic local ring of

L. 8.1.2. Zero-dimensional valuations of rank 2. Let rl be the nont.rivial isolated subgroup of r, with corresponding prime idea.l p C V. The composition of v with the order-preserving hOIIlomorphism r -+ r /rl defines a valuation II] or TJ of rank 1. Let Vi be the valuation ring of VI. V1 consists of the elements of [( whose value by v is in l' 1 01' is positive. Hence

8. Local Uniformization and Resolution of Surfaces

136

VI = "p. 11 illduce~ a valuation 112 of K(Vp ) with value group r 1 , where v2(f) = v(l·) if f· E Vp is a lift of f to Vp . Since K(Vp ) has a non-trivial valuation, which vanishes on K, K(v;.l) cannot be an a.lgebra.ic exLension or K. Thu~ !{(Vp ) has positive transcendence degree over !{ ,so it must have transcendence degree 1. Let x· E K{Vp ) be such that x* is transcendental over K. IJel.:r. he flo lin. of :c* to ~/' Then K(x) c Vp , so that VI is trivial on ]{(x), and 111 is a valuation of the one,.dimensional al~ebraic function field Lover j((x). Let V2 C K(Vp) be the valuation ring of 112, and let 1r : Vp ---+ K(v~) be the residue map. Then V = 7r- 1 (V2). VI is a I-dimensional valuation of L, so Vp is an algebraic local ring which is a local Dedekind domain. The residue field K(Vp ) is an algebraic function field of dimension lover K, so V2 is also a local Dedekind domain. Thus is a discrete group. Let w be a regular parameter in ~}'

r

Let'W be a regular parameter in Vp . Given any non-zero element f E L, we have a unique expression f = 111m. g , where nl, = Vl (/) E Z, and 9 is a unit. in Vp . Let.!/ he t.he residue of 9 in !«vp). Let n = 112(g*) E Z. Then, up to an order-preservin~isomorphism of r, zl(f) - (m, n) E r = Z2, where 712 has the lexicographic order.

8.1.3. Discrete zero-dimensional valuations of rank 1. Let x E L be such that 11(X) = 1. If y ELand v(y) == nl, then there exists a unique 1~1 E K Rtlch t.hat. '11,2 -= v(y - cp:n 1 ) > v(y). If we iterate thh; COIL~true­ tion infinitely many times, we have a uniquely determined Laurent series expanslOn CIX 'ftl

+ C2X + ... 1t2

with v(y - CIX n1 - . . . - cmX n ",) = nm+l > n m for all »'1,. This construction defines an embedding of K-algebras 1.J C K((:l:)), where K((:J.:)) is the field of Laurent ~eries in x. The order valuation on K((x)) restricts to the valuation v on L.

8.1.4. Non-discrete zero-dimensional valuations of rank 1. We subdivide this case by the rational rank of 1I. The rational rank is the dimension of r ® Q a.., a rational vector space. Sinee v has rank 1, we may ~~Ulne that r c JR. If 1 ft r, we can choose 0 < ,,\ E r and define an order-preserving group isonl0rphism if> : r -. ~ r by ¢( n) = ~. Then v = ~~ 0 V ifi R. valuat.ioll or r, with Ute same valua.tion ring

{f

E

D I v(f)

> O} - {f ELI v(f) > O}

= V

as v. We may thus normalize v (by replacing V wit.h the equiva.lent va.luat,jon 1/) ~o that. r is an ordered subgroup of lR. and 1 E r.

8.2. Local uniformiz8.tion of a.lgebraic function fields uf surfaces

137

Suppose that 1/ hfts rational rank larp;er that 1. Choose x, y E L such that £I(X) = 1 and lI(Y) = T is a pm~itive irrational number. Suppose that F(x, y) = L ai.ixiy:i is a polynomial in :c and y with coefficients in !<. Then v(aijXiyj) = i + jT 1'01' any term with aij =f; 0, so all the monomials in the expansion of F have distinct values. It follows that v(F) = min{i + Tj I aij =1= O}. We r.oncludc that x and y are algebraically independent, since a relation F(x, y) = 0 has inlhlite value. Furthermore 1 if L 1 = K (:1:, y) and 1/1 = /,II £1, we have that the value group 1', of v} is Z+Z7'. Since r/r 1 is a finite group, we have that r is a group of the same type. In particular, r has rational rank 2. Now snppose that v has rational rank 1. We can normalize r so that it is an ordered subgronp of 0, whose denominators are not bounded, as r is not discrete. In Example 3, Section 15, Chapter VI [92], examples are given of two-dimensional algebraic function Helds with value group equal to any given subgroup uf the rationals. There is a nice interpretation of this kind of valuation in terms of certaiJl "formal" Puiseux series (cf. 11aeLane and Shilling [64]) where the denominators uf the rational exponents in the Puiseux series expansiun are not bounded.

Exercise 8.1. Let K he a field, R quotient field of R. 1. Show that the rule ll(X) which satisfies

=

.=

!<[x, y] a polynomial ring and L the

1, 1J(Y)

= 7r

v(f) = min{i +.17r I ~j if f =

Eai.j:r,i'yj E

defines a K-valuation on L

=1=

O}

K[x,y].

2. CODlpute the rank, rational rank and value group ~t

r

of v.

Let V be the valuatiun ring of v. Suppose that 7' E r. Shuw that.

IT = {j' E V I v(f) > 7'} is an ideal of V.

4. Show that. V is not a Noetherian ring.

8.2. Local uniformization of algebraic function fields of surfaces In this section, we prove the following local uniformization theOI·ern.

Theorem 8.2. Suppost'. Ilwt K is an algebraically closed jield of characteri.'itic zero, L is a two-dimensional algeb1uic junction field over K and 1/ is a valuation of L. 'Then there exists a regular' lO(~Q.l ri.ng A 1lJhich is essentially of finite type over K 'With quotient field L such that 1J domina.te.'1 A.

8. Lucal Uniformization and Resolution of Surfaces

138

Our proof is by a ca."e by case analysis of the different types of valuations of L whieh were classified in the previous section. If 1/ is I-dimensional, we have that R = V satisfies the conclusions or the theorerIl. We must prove the theorem in the ca."e whell v has dimension zero, and one of the following three cases hold: II has rational rank 1, or 1/ has rank 1 and rational rank 1, or v has rank 2. The ca.-:;e when II has rank 1 includes the cases when v is discrete and 1/ is not discrete. We will follow the notation of the preceeding section. Suppose that K[;r,], ... ,:Z:n] is a. polynomial ring, and rn - (Xt, ... , Xn). For f E K[:l:l, ... , xnL ord(f) will denote the order of f in K[XI,"" :1: n]m (ef. Section A.4). In particular, ord(f) = '1' if and only if f E m r and f ¢ m r +1 • 8.2.1. Valuations of rational rank 1 and dimension O. Lelllma 8.3. Suppose that S = Krxll ... , Xn] is a polynorninl dng, II is a rank 1 valuation with valuation ring V and maximal ideal mv which contains S such. lhat.K(V) = K and mv n S = (Xt, ... , x n ). Suppose that there e:l;isl fi E S for i E N such that

1/(fl) < v(h) < ... < I/(fn) < ... 'l,Ci

.11 .9t1 tetly N

increasing sequence. Then limi_oc v(Ji) =

00.

Proof. Suppose that p > 0 is a positive real munber. We will show that there are only finitely IIlany real numbers less than p which are the values of elements of S. Let 7 = min{v(f) I f E (XlI'" ,:r:n )}· T is well defined since S is Noetheria.n. In fad, if there were not a minimum, we would have an infinite deseemling sequence of positive real numbers, 71

and clementfl 9i

>

E (:l~ 1, ... , :.l:n)

I Tr

-

79~

> .. . To'1. > ...

such that v (Ui) =

{f

E

S I v(f) >

The Jr.. are distinct ideals of 8 whieh

('OrIn

7i.

Let

Til·

a strictly ascending chain

IT1 cIT2 Coo·cIT•

C···

I

which is impossible. Choose a posi tive integer r such that r > ~. If f E S is such that 1/(f) < p, then write f = g + h with h E (Xl, ... , :Z;nt a.nd deg(g) < r. v(h) > p implies 1/(9) = 1/(f). Thus there exist a finite number of clements 91, ... ,9m E S (the monomials of degree < r) such that if 0 < A < P is the value of an element of S, then there exist Cl I • . . ,em E K such that (8.2)

V(qgl

+ ... + Cm 9m) = A.

8.2. Local uniformization of algebra.ic function fields of surfaces

139

We will replace the gi with appropriate linear combinations of the gi so

that

V(gl) < V(92) < ... < v(gm). Then A = V(9k) in (8.2), where Ck is the first non-zero coefficient.

By induction, it suffices to make a linear

chan~e

in the gi so that

(8.3) Aft.er reindexing the gi, we can suppose that there exists an integer l > 1

such that and

V(gi) > v(al) if l < 'i.. The equality 1/( gl JJi.) = 0 for 2 < i -< l implies fli E V and there exist !Jl Ci E K(V) = K such that fJ.i has residue Ci in K(V). Then JJi. - Ci E mv Y1 ~ implies v(~ - Ci) > 0, so that V(gi- Gig1) > V(gl) for 2 < i < 1. After replacing gi with gi-- Gig1 for 2 < i < l, we have that (8.a) is satisfied. D

Theorem 8.4. Suppose that S = K[x, y] is a polynomial rin.q over an algebraically closed field K of characteristic zero, 1) is a rational rank 1 valuation of K(x, 'V) with valuation "wing V and maxi.mal icleal rnv, K(V) = V/m.v = K such that S c V and the center of V on S is the maximal idea,l (x, y). Further suppose that f E K[x, y] is given. Then there exists a birational extension K[x, y] --t /([:1:/, y/] (/([:1:, 'V] and, /([:1;', y/] ha'/U·~ a CO'1mnon quotient field) such that K[x', y/] C V, the center of v on ]{[x/, y/] is (x/, y/), and f = (x/)lJ, where l is a non-negative integer and 8 E K[x', y1 is not in

(x/, y/). Proof. Set r = ord 1(0, y). Vole have 0 < r suppose that r > O. We have an expansion

<

00.

=

If T

0 we are done, so

d

(8.4)

f

=

L XQi'Yi(X)yt'i + LX i=l

Oi

/'i(X)yi3i,

i>d

where the first sum is over terms with minimal value p = V(X°'i"Yi(X)Y.8,) for 1 < i < d, 'Yi(:l~) E K[x] are polynomials with non-~ero COl1Rtant term, 131 < ... < 13d' 13d+l < t3d+2 < ... and V(XQ;'/'i(X)Y.8,) > P if i > d. Q I/(X) = ab Wit . J-I U,} - l- E W - . eX18 . -'I.. non-nega,lv~ t" t ~:le~• II(Y) J~, ( U, b) =1 . Tl - tele 1l1.egers

u/, b' sueh that ab' - bu' = 1. TheIl l/( ~) = 0 implies tJlere existfol 0 =F C E u K(V) = K such that v(~ - c) > O. Set x X

= Xla( Yl

+ C)a

f

,

Y = Xlb(Yl

f

+ C)b .

140

8. Local Uniformization and Resolution of Surfac,€s

We have that 'b a = X b' Y-, a Y1 + r, = X- Y

Xl

so that V(U1)

> 0 and

IJ~) lab' _ a'b]

V(Xl) =

Set a = Q1a + .BIb. We have that exist Ci E Z such that (

for 1 < i

< d.

D:ia

Qi -

a'

v~) > O.

=

+ l3ib =

b) ( b' l3i - {11

a

( 1 )

-

for 1

0

< i < d. Thus there

(0) e.i

By Cramer's rule, we have

f3i - th = Det Thus ei

I

(a,a

0 ) =

ei

aei.

= tJi~{b.

We have a factorization

f

= :l;(l It (Xl, Yl)' where tl

.f1 (Xl, Yl) =

(Yl

+ c)a'Q:1 +b' 81 (2: 1'i(YI + c) P, :i'i + :Z:10 1

)

i=l

is the strict transform of 1 in the birational extension K[Xl' Yl] of Kf:v, yJ. We have '1'1 < 00. ord 1(0, y) = r implies there exists an i such that Qi = 0, 13i = r, Thus ;)(yT) > p, and Pi < r for 1 ~ i < d. We have f3d - (31 < T. a If Tl < r, then we have a reduction, and the theorem follows from indlletion 011 T, sinee the conclusions of the theorem hold in K[x, yJ if r = O. We thus assume that Tl = 1'. Then I3d = r, /31 = 0, a = 1 and Ud = O. Further, there 1'1=

.

ord it (0, yt} <

is an expression d

L: l'i(O)(Yl + c)~\ = I'd(O)yr· i=1

Let

'f/'

be an indeterminate, and set cl

g(u) =

2: l'i(O)u)3i. = rd(O)(U - cr· i=l

The binomial theorem implies that (3d-1

= T - 1., a.nd thus

v(y) = V(XQ:d-1).

Let 0 i AE k(V) = k be the r~~idue of ;nat!. Then v(y - AXOd - 1 ) > v(y). Set y' = Y - AXOd - 1 and I'(x, y') = I(x, y). We have ord I' (Ot y') = ord/(O, y)

= r.

8.2. Local rmiformization of algebraic fUTlctioll Helds of surfaces •

141

y1' is a minimal value term of I(x, y) in t.he expansion (8.4), so 1I(Y)

<

11(/)· Replacing y with y' and f with I' and iterating the above pror.edure after (8.4), either we ar.hieve a reuuctiou ord/I(O,Yl) < Tj or we find that there exists a/ex) E K[x] such that v(y' - a'(x)) > v(y'). We further have that v(y') < v(f). Set y" = y' - a'(x) and repeat the algorithm fo!Jowing (8.4) until we either reaeh a reduction ord!I(O,Yl) < T or show that there exist polynomials a(i)(x) E K[x] for i E N such that we have an infinite increa..~ing hounded sequence

v(a(x)) < v(a'(x)) < ., .. Since each v(a(i) (x)) is an integral multiple of v(x), this is impossible. Thus there is a polynomial q(a:) E K[~l:] sud. that after replacing Y with Y q(x), and following the algorithm after (8.4), we achieve a reduction rt < r in K[xt, Yl]. By induction Qn T, we can achieve the conclusions oj' the theor·em. 0 We now give the proof of Theorem 8.2 in the case of this subsection, that is, v has dimension 0 and rational rank 1. Let x, y E L be algebraically independent over 1< and of positive value. Let w be a primitive element of Lover K(x, y). We can assume that w has positive value, for if v( w) = 0, there exists c E K such that t.he residue of 'l/} c is non-zero in K(V) = K, and we can replace w with w - c. Let R = K[x, y, w]. Then ReV, the center of v on R is the maximal ideal (x, Y, w), and R has quotient field L. Let z he algebra.ically independent over K(:J.:, y), and let I(x, y, z) in the polynomial ring K[x, y, zj be the irreducible polynomial such that R "" K[x, y, zJ/(f) (with w mapping to z + (f)). After possibly making a change of variables, repla.cing :z; with x + Ijz and y with Y + 7 z for suitable (3, 7 E K, we may assume that l'

= ord f(O, 0, z) <

(X).

If r = 1, then RmvnR. is a regular local ring which L~ dominat.ed by v (by Exercise 3.4), so the conc1usiolll:l of the theorem are satisfied. We thus suppose that r > 1. Let 7r : Klx, y, zJ --+ K[x, y, w] be the natural surjection. For 9 E K[x, y, z], we will denote v(rr(g)) by v(g). Write d

I(x,y,z) = La'i(x,y)ZtTi

(8.5)

i=l

+ Lcti(X,y)ZGi, i>d

where Ui(X, y) E K[x, y] for all i, the ai(x, y)z P for i > d and 0"1

<

0"2

< ... <

(J"d.

142

8, Local Urli[oClnization and Resolution uf Surfaces

By Theorem 8,4, there exists a birational ext.ension K[:l:, y] with ]([XI, YI] C V and mv n K[XI, YI] = (Xl, YI) such that

---t

K[:l:l, Yl]

d

f

= ~ X~"(li{Xl,

ydz l1i + ~ x~1ai(xl' Yl)Z/J'i,

i=l

i>d

where Ui(O, 0) =I 0 for all i. HI = K[xl, Yl, Zl]/(!) is a domain with quot.ient field L, R -+ R I is birational, R 1 C V and mv n R I = (Xl, Yl, z). Write t re It ' a 'IveI y prIme POSI't"Ive integers such that s't - st' = 1. Set v(z) -V(Xl)

t8 WI'th

'1' ,,1

09,

=

t

'r•• ''''2

{~42

+ t'·~2, )t l

Y1

= Y2,

L e t..~J "t J I)e posl.lve 't'

In t egers.

l,ll (z 2 = :'2

Z

+ C'·2 )Sl ,

where 0 =F C2 E K is determined by the condition V(Z2) > O. Let a = Alt + al'~' We have a = Ait + a'is for 1 ~ i < d. We have (as in t.he proof of Theorem 8.1) that Oi~Ol is a non-negative integer for 1 < i < d and 1 = X2 12($2, Y2, Z2), where (l

f2(XZ, Y2, Z2) = (X2

+ C2)tIAl+S'0'1(~ ~(Z2 + C2) O'i~171 ) + X2n i-I

is the strict trallsfonn of f in k[X2' Y2, Z2]. Set R2 = K[:J~2, Y2, Z2]/(!2). Then R 1 ~ R 2 is birational, R2 C V and mv n R'l. = (X'l, Y2,Z2). Set

rl If rl

.

= ord 12{O, 0, Z2)

<

(J'd -

t

(J'l

< r.

<

r, we have a reduction, so we may assume that rl = r. Then we have (J'd = r, (J'l = 0, t = 1 and ad{O, 0) =F 0. We further have a relation d

~ ai(O, 0)(Z2 + C2)U" = Ud(O, 0)z2' i=l

Set d

g(u)

= ~ ai(O, O)uO'~ = ad(O, O){u -

C2)T.

i=l

The binomial theorem shows that (J'd-l = r - 1 and v{z) = v(ad-l(x, y)). 'rhere exists c E k such that v(z - catl-1 (:1;, V)) > v(z). Then ru,dz.,.-l is a minimal value term of so that v{U) > v{z). Set i = z- ca(t-J(x, y),

U,

f'{x, y, zJ)

= f{x, y, z).

We have ord(f' (0, 0, i))

= %f. f replaced by !'

= r and

~

We repeat the analysis following (8.5)~ with replaced with z'. Either we get a reduction rl = ord{h(O, 0, Z2)) there exists a/(x, y) E K[x, y] f411ch that v{z' - a.' (x, y)) > v(zJ) and

v(z') < veal' ) = v(aJ). 8z' 8z

and z < r, or

8.2. Local ,mifol'wization .of algebraic function fields of surfaces

143

Setz" = z'-a'(x, y) and repeat the algorithm following (8.5). if we do reach a reduction Tl < r after a finite number of iterations, there exist polynomials a(i)(x, y) E K[x, y] for i E N such that v(a(x, y))

< V(ll'(X, y)) < ...

is an increasing bounded sequenee of real numbers. By Lemma 8.3 this is impossible. Thus there exists a polynomial a(x, y) E K[x, y] such that after replacing z withz - a(x, y), the algorithm following (8.5) results in a reduction '1'1 < 'r. We then repea.t the algorithm, so that by induction on "', we COIll:ltruct a birational extension of local ring~ R ~ Rl so that R I is a regular local ring which is dominated by V.

8.2.2. Valuations of rank 1, rational rank 2 and dimension O. After

normalizing IJ, we have that its value group is Z + ZT, whereT is a positive irrational nmnher. Choose x, y E L such that v(x) = 1 and v(y) = T. By the a,llalysil:l of Subsection 8.1.4, x and yare algebraica.lly independent over K. Let w be a primitive element. of Lover I«x, y) which has positive value. There exist.s an irreducible polynomial f(x, y, z) in the polynomial ring K[x, y, z] such that T = K~, y, w] f'¥ K[:l:, y, z] /(f) (where w maps to z+(f)) and mvnT = (x, y, 'W). ArLer posl:libly replacing x with x +t3z m and y with y I ,i"', whel'e m. i~ sufficiently larf'?;e that mv(w) > lnax{v(:c), v(y)}, and (J, l{ are suitably chosen, we nlay assume that

,E

r = ord(f(O, 0, z)) <

00.

If 'I' = 1, T:rnvnT is a re~ular local ring (by Exercise 3.4), 1:10 we assume that r > 1. Let 1f : K[x, y, zJ --+ K[:r., y, w] be the natural surjection. For g E I([x, y, z], we will denote l/(1r(g)) by IJ(q). Write d

f =L

ai xQi y/3i z 11

+L

a'ixOiy{J; z'"Yi,

£>Il

i=l

where all ai E K are non-zero, (J,i:1:Ui y/3i Z"Yi for 1 value terms, and 11 < 12 < ... < 'Yd·

< i < d

are the minimal

There exist integers s, t such that v(z) = .~+Tt, and there exist integers .A-f, N such that lJ(XQ(y/3~Z"ri) = M + NT for 1

(8.6)

S i < d. We thus ha.ve equalities M Q'l + S"YI = N

= /31 + til

=

=

ad

+ .'l,d,

= fJd +t"Yd'

V,le further have (Hi

+ s,i) + (/3i + t"Yi)T > M + NT

8. Local UnifonmzatioIl aIld Resolution of Surfaces

144

> d.

Since ord(j(O, 0, z)) = l' and (J,d;r.:Ud y i3d z''Id is a minimal value term, we have 'Yd < r. We expand T into a continued fraction:

if i

1.

7"= h1+

11,2

+

1 h3+".

Let ~ be the convergent fractions of 7" (cf. Section 10.2 [46]). Since linl £i. ~. ~ i , we have

=

+ N lJ gjgj

(ai + S1i) + U3i + t'Yi)ll > M

1<.)1' j -= l' - 1 and j = p, whe~leveri > d and p is sufficiently large. We further have that sgp+tjp > 0 and syP 1 +tlp-l > for 1) sufficiently large. Since v(x'<;yt) = v(z), there exists e E [( = K(V) such that IJC;;yt - e) > O. Consider the extension

°

(8.7) where 'l.gp 'y9 p- l . '1 1 ,

(8.8)

X"!'P'yfp-l 1 ,

y Z

x'<;yt(Zl

We have fp-lg p - !p.9p-l

.

= X~9P+t!Py:9p-l itfr -l (Zl

+ c)

+ c).

(ef. Theorem 150 [46]), and €, -T + ;~_~

= € = ±l

and". - b. have t.he same Rjgm~ (cf. Theorem 154 [46]). From (8.8), we see 9p that X!P-l

X·l!l--

yOP

-, y9,,-1 ' YI- :r.Jp from which we deduce that (8.7) is birational, and E1J(Xl)

= Yp

E·

1

1(111-

i),

-

tV(Yl) = Op(r _ fp) . 9p

•Q p-l

> O. We have

Hence V(Xl), V(YI), V(Zl) are all

N .,. .) , f( x, y, "".,) -- x tAlgp +Nf'P YlMg p-1+ jp-1 .f1 (Xt, Yb ""1

where

d,

it = (Zl +

err (L 1

ai(Zl

+ e)ii

11)

+ $IYIH

i-I

is the strict. t.rallsfornl of f in K[xt, Yt, Zl]. We thus have constructed a birational extension T ~ T 1 = Krxl, Yb zl]!(ll) sueh t.hat. 1'1 c V and

mv n Tl

= (Xll Yl, zI).

We have

It (0,0,0) =

0,

80

'lJ,

= r, is a root of d

rjJ(u) =

L:CI.{(f,"Ii-'Yl

i=l

= O.

8.2. Local llIlilhrmization of a.lgebraic function fields of surface~

145

Let 'f'1 = ordft(O,O,z1)' Then rl < ""Id - ""Ii < T. If 'rl < 'f', we have obtained a reduction, so we ma.y assume that rl = r. Thus '"Y1 = 0, 'Yd = T, ad = t3d = 0 (since '"Yi < r for 1 < i < d) and

¢(u) Hence '"Yd-l

= r - 1 and

= ad(u -

ct.

ad-l =1= 0, by lhe binomial theorem.

We have v(z) = CY.d-l +/ld-lT. Our conclusion is that there exist non-negative int.egers m and n such that IJ(Z) -"'1, + nT. There exists c E K such that

lI(z -

(~;f,myn)

> v(z).

We make a change ofvariahles in K[:l:, y, z], replacing z: with z' = z - cxmytl, and let f'(x,y, z').:..:. f(x,y,.z). We have l,1(Z') > v(z) and ordf'(O,O,z') = r. Since radZr-1 is a minimal value term of we have v(z) < v(U). '\Ve

M,

further have ~ = ~!;. \Ve now apply a t.ransrormation of the kind (8.8) with respect t.o the new variables Xl Y, z', If we do not achieve a reduetion 'I'I < '1', we have a relation 1,1(Z') = 1n1 + nl T with 1n" n, non-negative inte~ers, and we have

8f

v(z) < v(z') < v( 8z)' We have that there exists

C2

E K such that

v(z' -

C2 XmJ

yn J ) > l/(z').

Set z" = z' - C2.r,'Ul 1 y11.] .We now iterate the procedure starting with the transformation (8.8). If we do Tlot a(~hieve a reduction Tl < r after a finite number of iterations, we eonst.rnct an infinite bounded sequence (since ~ ! () in 1') rn+nT < m'l +nlT < ... < m.i+ niT < ... , where mi, ni are non-nep;ative integers, which is impossible, We thus must achieve a, reduction rl < r after a finite number of interations. Bv induetion '" on r, we can construct a hira.lional extension T -.> 1'1 such that Tl C V and ('1; )mvnTl is a regular local rinp;. 8.2.3. Valuations of rank 2 and dimension O. SUPPosP. that lJ has rank 2 and dimension n. Let 0 c p C m'l be the distinct prime ideals of V. Lp.t. x, y E L be such that X, yare a transcendence basis of Lover I(, lI(x) and v(y) are positive and the residue of x in [((Vp ) is a t.ransCendp.llce basis of I«(Vp ) over I{. Let 'W be a primitive element of Lover K(x, y) such that v(w) > o. Let T = Klx, y, w] C V. Let !(:1:, y, z) he an irreducible clement in the polynOluial ring K[x, y, z] such that

(8.9)

I([x, y, z]/(f)

where z + (f) is mapped to w.

rv

Klx, y, w],

146

8. Local Uniformization and Resolution of Surfaces

By our construction, the quotient field of T is Land trdegK(T jp n T) = 1. Thus p n 'T is a prirne ideal of a curve on the surface spec(T). Let Co be the curve in sper.(T) with ideal p n Ro in Ro. Corollary 4.4 implies t.here exist.s a pruper birational morphism 7r : X --) spec(T) whieh is a sequence of blow-ups of points such that the strict transform C u of Co on X is non-singular. Let Xl ~ X be the blow-up of Co. Since Xl . ~ spec(T) is proper, there exists a unique point al E Xl such t.hat V dominates Rl = 0 Xl,al' Rl is a quotient of a 3-dimensional regular local rinK (as explained in Lemma 5.4). Let. C1 be the curve in Xl with ideal p n R] in R l • (Rt)pnRl dominates RpnR and (Rdpnnl is a local ring of a point all the blow-up of the maximal ideal of R17n R.. We ~all it.erat.e this procedure to construct a sequence of local rings (with quotient field L) Set

Ro

=

Rm v nl"

-

Ro

~

RI

--+ •.. --+

Ri

-t ...

such that V dominates Rifor all i, Ii;, is a quotient of a regular local ring of dimension 3 and (R;,idpnRi+l is a local rin~ of the blow-up of the maximal ideal of (R)pnRi for all i. Furthermore. each Ri is a lor.alhm.l,iou of a quotient of a polynomial ring in :i variables. Sin~e

(Ro)pnRo can be considered as a local ring of a point on a plane eurve over the field I«(x) (as is explained in the proof of Lemma 5.10), by Theorem 3.15 (or Corollary 4.4) t.here exists an i such that (lli)pr,~ is a regular local ring. After replacing the T in (B.B) wit.h a slli1.able a.ftine ring, we can now assume that T~nT is a regular local ring. If TmvnT is a regular local ring we are done, so suppose T mv nT is not a regular local ring. Let. 1) = 7r I (p n T) 1 where 1T' : K[x, y, z] ~ T is the natural surjeet.ioIL. If 9 E filx, y, zj is a polynomial, we will denol,e v(7r(g)) by II(g). Let I C I([x,y,z] be the ideal I -= (f, ~) defining the singular locus of T. Since I ct p (as TpnT is

U. U,

a regular local rin~), one of 7r( g£), 1r(U), 7r(

.

M)

8f

of

of

X

Y

.z

¢]J

n T. Thus

mm{lI( -8 ),v( -8 ), v( 8·- )} = (0, '11.) for some n E N.

~rrite

f = F~, + F

7'

11

+ ... + Fm :

where Pi is a forIII of dep;ree i in x, y, z and r = ord(f). After possibly reindexin~ the x, y, z.' we lllay assume that 0 < vex) < II(Y) < lI(z). Since K (V) - K, there exist c, d E k (which could be 0) such that

v(y - ex) > v(x) and v(z - elx)

> v(x).

8.2. Local unifol'ImzatioIl of algebraic [UIlctioIl fields oJ'slld'a.ces

Thus we have a birational transformation K[x, y, z] ~ K[x], Yh

147

ztl

(a qua-

dratic transformation), where

z- dx x

Z1 = - - - ,

wit.h inverse

h (Xl, Yll Z1)

=

+ C, Zl + d) + :V1 F,'+I (1, YI + G, Z1 + d) +, .. + x¥"-'''Fm(l, Yl + C, Zl + d) F1·(1, Yl

Hence

,

8j

8j

81

uX

vy

uZ

nlln{v(~),v(:~),v(~)}

.

8ft

> (r -1)V(Xl) +ffiln{v(a

:r.1

8il

8ft

Y1

ZI

),v(8~),v(a-;:--)}'

Since we have assumed that Tm vnl' is not regular, we have r > 1, We further have that v(X!) > 0, so we conclude that

Let rl = ord(h)

< r, (TI )mv nTl is a

Thus we may assume that

fl

l'ef,'1.1Iar local ring if and only if'1'1

= 1.

> 1.

We now apply to 1, a new quadratic transform, to get

8j 8f 8j' min{11 «(} 2), v «(} 2), v «(} 2)} = (0, n2) X2

Y2

<

(0, n I) .

Z2

Thus after a finite number of quadratic transforms T ....... 11, we construct a Tt such that (Tt )mvnll is a regular local ring which is dOlninated by V.

Remark 8.5. Zariski proves local uniformization for arbitrary characteristic zero algebraic function fields in [87].

148

8. Local Unifornlization and Resolution of Surfaces

8.3. Resolving systems and the Zariski-Riemann manifold Let L/K be an algebraic function field. A projective model of L is a projective K -variety whose function field is L. Let E(L) he the set of valuation rings of l~. If X is a projective rnodel of L, then there is a natural morphism 1rx : E(L) -+ X defined by 1rx (V) = p if p is the (unique) point of X whose local ring is dominated by V. We will say t. hat. IJ is the cell tel" of V 011 X. There is a topology OIl E (L) , where a basis for the topology are the open sets 1r X l (U) for an open set U of a projective model X of L. Zariski shows that E(L) is quasi-compact (every open cover has a. fillite 8ubeover) in [89} and Section 17, Chapter VI [92]. E(L) is called the Zariski-Riemann manifold of L. Let N he a set of zero-dimensional vaillt ion ringfol of D. A resol v j ng system of Iv is a finite collection {81 , ... , Sr} of projective models of L such that for each V EN, the center of V 011 at least one of the Si is a non-singular point.

Theorem 8.6. Suppose that L is a two-dimensional algebraic function field over an algebraically closcd ficld K of characteristic zero. Then there exists a 'l'f:,~ol'lJing system, f(}',. the set, of aU zero-dimensional valuation 1'ings of L. Proof. By local uniformization (Theorem 8.2), for each valuation W of L there exists a projective model XHl of L such that the center of W on Xw is non-singular. Let Uw be an open neighborhood of the cent.er of \.1.1 in X w which is non-singular, and let Aw = 1r WI (Uw), an open neighborhood of W in E(L). The set {Aw I W E E(L)} is an open cover of E(L). Since ~(L) is quasi-r:ompad., there is a finite Ruhcovp,r {Awl' ... ' Aw,.} of E(D). Then X ,"V!, ... I",Y\Vr is a resolving system for the set of all zero-dimensional valuation rings of L. D Lenun8 8.7. SUP1JO,'1f: /,Iud L

';','4 lJ.

I:wo-di."u:';'fu,imwl alge.lnuic fll:ncl'i.on

Jieltl

ove1' an algebraically closed field !( of characteristic zero, and

flo

C Rl C ... C Ri C ...

is a sequence of distinct normal two-dimensional algebm,ic local rings of L. Lf~t. n = U R'i' If eVf,;'ry one-dim.ensional 'valuation Ting V of L which contains n intersects Ri in a hei.qht one prime for some i then n is the I

valuation ring of a zcro-dim.en.r,ional valuation of L. Proof. n is by eOJlstruetiuIl integrally dosed in L. Thus n is the interHectioll of the valuation rin~s of L containing n (d. the corollary to Theorem 8, Section 5, Chapter VI [92]). Observe that n '# L. We see this as follows. If 'ffl,i is the maximal ideal of ~, then m = Umi is a non-zero ideal of n. Suppose that 1 Em. Then 1 E mi for some i, which is impossible.

8.3. Resolving systems and the Zariski-R.iema.lln manifold

149

n

We will show that is an intersection of zero-dimensional val uatioIl rings. To prove this, it, slIffices t.o show that if V is a one-dimensional valuation ring containing n~ then there exists a zero-dimensional valuation ring W such that V is a localization of Wand 11 c W. Since V is onedimensional, V is a. local Dedek i nd domain and is an algebraic local ring of L. Let K(V) be the residue field of V. Let 1nv be the maximal ideal of V, and let Pi = R;. n mv. By asSUlllptioll, there exists all iudex j such that i > j implies that IJ·i, is a height one prime in Ri. (Ri)Pi is a normal local ring which is dominated by V. Since both (RdPi and V are local Dedekind domains, and they have the COllllllon qnotient. lield L, we have that (Hi) Pi =- V ror i > j (ef. Theorern 9 ~ Section 5 ~ Chapter VI [92]). Thus the residue field Ki of (Ri)Pl is !«(V) for i > j. The one-dimensional algebraic local rings 8 i = Ri/Pi for ·i > j have quot.ient field [{(V), and Bi +I dominat.es Si. USi is not K(V) by the same argument we used to show that is not L. Thus there exists a valuation ring V' of K(V) which dominat.es UBi. Let 7r : V -----} K(V) be t.he residue map. Then W = rr-1(V') is a ~ero-diIIlensional valuation ring of V and W contains n.

n

n

Since we have established that is the intersection 01" ~ero-di mensional i~ not contained in two distinct valuation rings, we need only show that zero-dimensional valuation rings. Suppose that is contained in two distind. zero-dimensional valuations, VI and V2 .

n

n

We will first. construct a projective surface Y with function field L such that Vi and V2 dominate (unique) distinct points ql and q2 of Y. Let ;1:, y be a transcendence basis of Lover K. Let VI, V2 be valuations of L whose respective valuation rings are VI and V:l- If 1/ 1(x) and 1I2(X) are both < 0, replace x with :~. Suppose that 1I1(X) > 0 and V2(X) < O. Let c E K be sueh that the residue of x in the residue lIeld K(V!) ~ !{ of VI is not c. After replacing x with x - c, we have that V1(X) = 0 and 1I2(X) < O. We can now replace x with ~ to get that lII(X) > 0 and 1I2(X) > n. In this way, aller possibly changing x and y~ we ean aSSUIue that J.: and yare contained in t.he valuation rings VI and V2 - Since VI and V2 are distinct zero-dimensional valuation rings, there exist f E V1 such that f r{. V2l and 9 E V2 such that g f/. Vi (cf. Theorem 3, Section 3, Chapter VI [92}). As above, we can assUllle that 111(1) = 0, v2(f) < 0, 1I1(,Q) < 0 and 1I2(g) = o. Let a = }, b = ;, and let A be the integral closure of Iqx, y, a, b] in L. Dy construction J A c Vi n V2. Let. rnl = A n 'mY" 'ffL2 = A n rnV2' We have that a E m2 but a ¢ mI, and b E ml but b ¢ m2. Thus ml :/= m2. We now let Y be the normalization of a projective closure of spee(A). Y has the desired properties. Since Ri is essentially of finite type over K, it is a localization of a finitely generated K-algebra B i which has quotieut field L. Let Xi be the

8, Local Uniformization and Resolution of Surfaces

150

normalization of a projective closure of spec(Bi ). Since.Rt. is integrally closed, the projective surface Xi has a point ai with associated local ring OXi,ai =

Hoi.

There is a natural birational map 11 : Xi ~ Y. We will now establish that Ti cannot be a morphism at ai. If Ii were a morphism at ai, there would be a uHique point bi Ull Y such that Ri = Oxt• u' dumina.tes Vv-, b .. Since Vi and V2 both dominate 0 Xt,atl Vi and V2 both dominate 0Yi1b. a contradiction. )t,

.It,,t

l

,

By Zariski's main theorem (cf. Theorem V.5.2 [47]) there exists a onedimensional valuation ring W of L such that W dominates OX.ita, and W dominate:;; the local l'ing of the generic point of a curve 011 Y. Let r,i, be the set of one-dimensional valuation rings W of 1< such that W dominates Xl ,Oi. and W dominates the local ring of the generic point of a curve on Y. r,t: is a finite set, since the valuation rings W or ri are in 1-1 conespondence with the curves contained in 1T11 (ai), where 1Tl : r 1 ,. --+ Xi is the projection of the graph of Ti onto Xi. Since Ri+ 1 = 0 Xil1 ,ai i1 dominates H1: = 0 Xi ,n., , we have riot"! c ri fur all ,i. Since each ri is a nUll-empty finite set, there exists a one-dimensional valuation rinp; W of L such that W dominates Ri for all i, a contradiction. 0

o

Theorenl 8.8. Suppose that L is a two-dime.ns·ional alge.brnic functiun field over an algebraically closed field K of characteristic zero, R is a two-dimensional normal algebraic local ring with quotient field L, and

i.9 a .9p.q7Lp.17.Ce of 1w'rmal local T'ing.f1 !J'llch thai, Ri +l 'i.fJ obtained from Ri by

blowing up the maximal ideal, normalizing, and localizing at a maximal ideal so that Rt,+l dominates Ri. Then n = U Rt, is a zero-dimensional valuation ring of D.

Proof. By Lemma 8.7, we are reduced to showing that there docs not exist

a one-dimensional valuatioJl ring V of E sud. that n is eont.ained in V and mv n ~ is the maximal ideal for all i. Suppose that V is a one-dimensional valuation ring of L which contains n. Let 1/ be a valuation of L whose associated valuation ring is V. Since V is one-dimensional, there exists 0: E V such that the residue of Q in K(V) is transcendental over K.

Let mi be the maximal ideal of Ri. Assume that mv n R is the maximal ideal mo of R. Write a = ~ with f., "7 E R. Then a rt R, since R/17~O K. Hence "7 E m.o, and also E rno since v(n) > O. Since Rl is a local ring of the normalization of the blow-up of mo~ moRl is a principal ideal. In fact if

e

I'V

8.:~.

R.esolving systems ftnd the Za.riski-RiemCillIl manifold

mo is such that v() = min{v(J) I J Thus 6 = ~ E R 1 and 1]1 = ~. E R1. ( E

If we assume that mv n Rl 6,772 E R2 and

=

E

m,o}, we have that (R I

ml, we have that ~I, 7]1

V(~)

> v(6) > 0, IJ(1]) > 1J(1]1) > 0,

151

( ill 1 ,

a

= mURI'

=

~ with

v(6) > V(~2), V(1]1)

> V(1]2).

More generally, if we assume that mi = mv n V for i = I, ... ,n, we can find elements {i, '1Ji E R;, for 1 ::; i ~ n such that a = ~ and v(~)

> V(~l) >

> IJ(~n) > 0,

> v(1]d >

> V(1]n) > O. Since the value group of V is IE, it follows that n < min {v(~), ~/( 7])}, Hence, v(1])

for all i sufficiently large, tnv n Ri is not. t,h~ maximal ideal rnv n ~ must t.hell be a height one prime in~, for otherwise we would have mvnRi = (0), which would imply that V = L, against our assumption. 0 R.emark 8.9. The eouclusions of this theorem are false in dimension 3 (Shannon [76]). Theorem 8.10. S''lLppmu~ Owl HJ s' aTe nonnal projective surfaces over an cLlgeb1'aically dosed field 1< of characteristic zero and T is a birational map from 5 to 5'. Let S 1 ~ 5 be the projective morphi.'Im defint~d by laking Uu: normalization of the blow-'ll.]} oj lhe finitely TTuLny points of T, where T is not a morphism. If the induced birational map Tl from 51 to Sf is not a morphism, let 52 ---+ 51 be the normalization of th.e blOW-'lI.l} oj lhe finitely many fundamental po·i.nt.., oj r i . We then iterate to produce a sequence of bir'ational morphisms of surfaces C' CI ' ..S ~ l:J 1 ~ ':J2 ~ • . . ~ D'L ~ • •• • (.1

This sequence must be of finite length, so that the inducrAf, Tational m.ap Si ~ S' is a morphism for all·i. .'Inffir.ienUy la'lye.

Proof. Since 5i is a normal surface, the set of points wher~ Tt is f10t a. lllorphism is finite (d. Lemma 5.1 [47]). Suppose that no T i : Si ~ S' is a morphism. Then there exists a sequence of points Pi E 5i for i E N such that Pi+l maps to Pi and Ti is not 0. Inorphism at Pi for all 'i.. Let Ri = OSi,Pi' The bil'at.iona.l maps give an identification of the function fields of the 5 i and Sf with a common field L. Dy Theorenl 8.8, n = UR,;. is a zerodimensional valuation ring. Hence t.heJ'e exists a point q E S' such that n domina.tes OS',q. OSI,q il:l a localization of a K-al~ebra B = ]([h,···, ir] for some il, ... ,iT E L. Since B c n, there exists SOllle i such that B c Ri,' Since n dominates R-;" we necessarily have that ~ dOlllinates OS',q· Thus Ii

n,

152

8. Local Uniformization and Resolution of Sllrla,ces

is a morphism at. p';., a. contradiction to our assumption. Thus our sequence 1I1U~t be finite. D

Theorem 8.11. Suppose that Sand S' arc projeetivp- .'I'll·rftL(;e~ over em algebraically closed field K of charaeter1..'ll'i(: zeT'O which form a resolving system for a .'1et of u~1·o-d'i.·rnensional valuations. N. Then there exists a projecti'/}e surface S* which is a resolving system for N . Proof. By Theorem 8.10, there exists a birational morphisln S - ~ 8 obtained by a sequence of normalizations of the blow-ups of" all point~ where the birational map to 8' is not. deriued. Now we construct a birational morphism S -lo S', applying the algorithm of Theorem 8.10 by only blowing up the points where the birational rnap to S iJ; not defined and which arc nonsingular points. The algorithm produces a biratioIlal Illap S' ---;'0 S which is a, morphism at all non-sinp;ular points of 8'. Let S* be the graph of the birational map from .9 to 8', wit.h natural projections 1r I : 05'" -lo Sand 7T'2 : S* --+ S'. We will show that 8* is 11 resolving system for N. Suppose that V EN. Let p, p', p' anu p* be the center::i of V on lohe re~pective projective surfaces S, S', S' and S*. First suppose that p' is a singular point.. Then p must be non-singular, ... _, ~., {Sl, S'} is a resolving system for lv. The birational map from .5' to 8 i~ a morphism at p, and 1Tl is an isomorphhmJ aL p. Thus the center of V on S* is a non-singular point. ~

Now suppose that p' is a non-singular point. Then fI is ft. nOIl-~illgular point of S' and t.he hirational map from S' to S is a morphism at ll. Thus 7r2 is an isomorphism at p', and the center of V 011 8* is a non-singular point. D

Theorem 8.12. Suppose ihnt L i:'1 lL two-dimensional algebraic function field O'1J~7' 1111. al.!Jt;b'l'lLiclLlly dosed field of characteristic zero. Then tlwn~ f.:r.'i8i.., (L non-singular pro.iective surface S with fund'ion jielcl L.

Proof. The theorem follows from Theorem 8.6 and Theorem 8.10, by induction on l' applied to a resolvinp; system {Sl, ... , 8 r } for the set or zeroD dimensional valuation rings of L.

Remark 8.13. Zariski proves the generalization of Theorem 8.11 to dimension :i in [90L and deduces resolution of sinp;ularities for characteristic zero 3-folds from his proof of local uniformization for charaeteristic zero alp;ebraic function fields [87]. Abhyankar proves local uniformization in dimension three and cha.racteristic p > 5, from which he deduces resolution of singularities for 3-folds of characteristic p > 5 [4].

8.3. Resolving systems aIld tile Za.";skj·Rjemann Dlnni[old

153

Exercise 8.14.

1. Prove tha.t. all the birational extensions constructed in Section 8.2 are products of blow-ups of points and non-~ing1l1ar cnrves.

2. Identify where characteristic zero is used in the proofs of this chapter. All l.mt one of the cases of Section 8.2 extend without great difficulty to characteristic p > O. SmIle eare is required t.o ensure that K(x, y)-_.~ L is separable. In [1] I Abhyankar accomplishes this and gives a very different. proof in the remaining case to prove local uniformization in characteristic p > 0 for algebraie 8nrfaees.

----Chapter 9 Ramification of Valuations and Simultaneous Resolution

Suppose that. IJ is an algebraic function field over a field K, and trdegJ< L < 00 is arbitrary. We will use the notation for valuation rings of Section 8.1. Suppose that R is a local ring contained in L. We will say that R is algebraic (or an algebraic local ring of L) if L is the quotient field of Rand R is essentially of finite type over K. Let K(S) denote the residue field of a ring S containingK with a unique rnaxiInal ideal. We will also denote the maximal ideal of a ring S containing a unique maximal ideal by ms. Suppose that L ---+ L* is a finite separable extension of algebraic function fields over K, and V'" is a. valuation ring of L* I K associated to a valuation v· with value group r*. Then the restriction II = 11* I L of 1/* to IJ is a valuation of L I K with valuation ring V = L n V·. Let be the value group of lI.

r

There is a commutative diagram

L

i

V = lJ n V*

155

9. R.nmificatioll of Valuations and Simultaneous Resolution

156

There are associated invariants (ef. page 53, Section 11, Chapter VI [92]) namely, the reduced ramification index of v* relative to 1I,

e = [r* : f] and the relative degree

or v* f

< 00,

with re:spect to

1/,

= [K(V*) : !«(V)].

The classical case is when V* is an algebraic local Dedekind domain. In this cfl.'3e V is also an algebraic local Dedekind domain. H 71l.V" = (y) and mv = (x), then there is a unit "Y E V· such that X

_ e -,y .

Since we thus have

r* jr '" ZjeZ, we see that e is the reduced ramification index.

Still assuming that V· is all algebraic local Dedekind domain, if we further assume that K = K(V*) is algebraically closed, and (e, char(K)) = 1, then we have that the induced homomorphism on completions with respect to the respective maximal ideals A

V

A

----t

V*

•

is given by the natural inclusion of K algebra.q

K[[x]]

~

K[[yJ],

.

where y = yPYy and x = It. We thus have a natural action of r* jr on V· (a generator multiplies y by a primitive e-th root of unity), and the ring of invariants by this action is

In this chapter we find analogs of these results for general valuation rings. The basic approach is to find algebrak local rings R of L and S of L + such that V* dominates 8 (8 is contained in V* and mv" n S = 1ns) and V dominates R, and such that the ramification theory of V ~ V· is captured ill

· R• ----t

(9.1)

£:.1 .:>,

and we have a theory for R ~ S comparable to that of the above analysis of V ._) in the specia.l ease when V* is an algebraic local Dedekind domain.

v·

This should be compared with Zariski's Local lJniformization Theorem [87] (also Section 2.5 and Chapter 8 of this book). It is proven in [87] that if char( K) = 0 and V is a valuation ring of an algebraic function field LjK, then V dominates an algebraic regular local ring of L The case when trdegK L = 2 is proven in Sect,ion 8.2.

9. R.amification of Valuatiun.s and Simultaneous Resolution

157

We first consider the following diagram:

L

~

L*

i

(9.2)

S

where L *I L is a finite separable extension of algebraic function fields over L, and S is a normal algebraic local ring of L *. We say that S lies above an algebraic local ring R of L if S is a localization at a nlaximal ideal of the integr al closure of R in L *. Abhyaukar and Heinzer have given examples of diagrams (9.2) where there does not exist a local ring R of L such that 8 lies above R (cf. [35]). To construct an extension of the kind (9.1), we must at least be able to find an 8 dominated by V· 8uch that S lies over an algebraic local ring R of L. To do this, we consider the concept of a monoida.l Lransfoflll. Suppose that. V is a valuation ring of the algebraic function field LI K, and suppose that R is an algebraic local ring of L such that V dominates R. Suppose that peR is a regular prime; that is, Rip is a regular local ring. If f E 11 is an element of minimal value, then R[7] is contained in V, and if Q = rnv n R[j], then Rl = R[r]Q is an algebraic local ring of L which is dominated by V. We say t.hat R ~ R 1 is a monoidal transform along V. Rj is the local ring of the blow-up of spec(R) at the non-singular sllbscheme V(P). If R is a regular local ring, thcn Rl is a regular local ring. In this case there exists a regular system of paraIIleters (XI, ... ,xn ) in R such that if ht(p) = 1', tlleu R 1 = R[i£2.: ..• , ;q h JQ. Xl The main tool we use for our analysis is t.he Local Monomialization Theorem 2.18. With the notation of (9.1) and (9.2), consider a diagram ~

L

i (9.3)

V

L*

i

= V* n lJ

~

V*

i S""

where S· is an algebraic local ring of L". The natural question to consider is local simultaneous resolut.ion; that. is, does there exist a diagram (constructed I'rorn (9.3))

V

~

V*

~

i S

T

R

i 8*

158

9. RaIIlification of Valuations and Simultaneous Resolution

sneh that 8* ~ S is a sequence of monoidaI transforms, S is a regular local ring, and there exists a re~ar local ring R such that S lies above R? The answer to this question is no, even when trdegJ((L*) = 2, as was shown by Abhyankal'ill [2]. However, it follows from a refinement in Theorem 4.8 [35J (strong monomialization) of Theorem 2.18 that if K has characteristic zero, trdegJ((L"') is arbit.rary and V* has rational rank 1, then local simultaneous resolution is true, since in this case (2.8) becomes Xl = (hy~ll, X2

= 112, ...

,Xn

=

Yn,

and S lies over Ro. The natural next question to eonsider is weak simultaneous local resolution; that. is, does t.here exist a diagrarn (constructed from (9.3)) V

~

i R

V*

i ~

8

i S· such that 8* ~ 8 is a sequenee of monoidal transfurms, 8 is a regular local ri ng, and there exist~ a normal local ring R such that S lies above R? This was conjectured by Abhyankar on page 144 of [8] (and is ilnplidt. in [1]). If trdegK(L*) = 2, the answer to this question is yes, as w~ shown by Abhyankar in [1]' [3]. IT trdegK(L*) is arbitrary, and K has characteristic zero, then the answer to this question is yes, as we prove in [29] and [35]. This is a simple eorollary of local mOllumialization. In fact, weak simultaneous local resolution follows from the following theorem, which will be of use in our analysis of ramification.

Theorem 9.1 (Theorem 4.2 [35]). [Jet K be a field of characteristic zero, L an algebraic function field over !(, L * a finite al.qebraic extension of L, and lJ* a valuation oj L· / !(, with valuation ring V*. Suppose that 8* 'i.'1 an algebraic local ring oj L * wh-inh is dominated by v*, and R* is an algebraic local ring oj L which is dominated by S*. Then there exists a commutative diagram 1l{J

(9.4)

~

R

c

V*

i R*

~

where S* ~ Sand R* ~ l1J.J are sequences of monoidal transforms along v* such that Ro ~ S have regular parameters of the form, oj the concl'IL:~iol1l:l of Theorem 2. 18, R is a n017nal algebT'aic local ring of L with tonc singularities

9. Ramification of Valuat.ions and SiIIlUltanoous Resolution

159

which is the localization of the blow-?11> of an ideal in Ro, and the regular lor:al ring S is the localization at a maximal ideal of the integral closure of R in L*.

Proof. By resolution of singularities [52] (d. Theorems 2.6 and 2.9 .in [26]), we first reduce to the case where R* and S" are regular, and then construct, by the Local MonomiaJization Theorem 2.18, a sequence of monoidal transforms along lJ*

Ro (9.5)

--+

c

V*

T

i R*

S

--+

S*

so that R(J is a. regular local ring with regular parameters (Xl, ... ,xn ), 8 is a regular local ring with regular parameters (Yl' ... , Yn), and there are units <5'1, •.• , 8n in 8, and a matrix of natural nUlllberl:l A = (aij) with non-zero determinant d such that

for 1 < i < n. After possibly reindexing the Yi, we may assume that d > O. Let (b.ij) be the adjoint Inatrix of A. Set

for 1 < i < n. Let R be the integral closure of Ro[fr, ... ,fnJ in L, localized at thfl center of v*. Since VmRS = ms, Zariski's Main Theorem (10.9 in [4]) shows that It is a normal algebraic local ring of L ~mch that S lies above

R. We thus have a sequence of the form (9.4).

D

Tn the theory of resolution of singularities a modified version of the wea.k simultaneous resolution conjecture is important. This is ealled global weak simultaneous resolution. Suppose that f : X --+ Y is a proper, generically finite morphism of k-varieties. Does there exist a commutative diagTaIIl

Xl

(9.6)

n.

1 X

-L

Yl

1 y

such that h is finite, Xl and YI arc complete k-varieties with Xl nOIlsingular, Y1 normal, and the vertical arrows are birationaJ?

160

9. Rawific8.tioll of Valuations and Simultaneous Resolution

Thisquestioll ha..l) been posed by Abhyankar (with the furt.her conditions that Y1 --+ Y is a sequence of blow-ups of non-singular subvarieties and Y, X are projective) explicitly on page 144 of [8] and implicitly in the paper

[2]. We answer this question in the negative, in an example constructed In Theorem 3.1 of [30]. The example is of a generically finite Jl1lorphism X ---10 Y of IlOIl-l5iIlgular, projective surfaces, defined over an algebraically closed field K of characteristic not equal to 2. This example also gives a r:onnterexanlple t.o the principle that a geometric statement. tha.t is true locally along all valuations should be true globally. We now state our main theorem on ramification of arbitrary valuations.

Theorem 9.2 (Theorem 6.1 [35]). Suppose that char(K) = 0 and we are given a diagram of the form of (9.8)

L

---10

i

L*

i

V = V* h L

-+

V*

i S" . Let K be an algebruic closure of K(V*). Then there exists a regular algebraic local ring Rl of L such that if Ro is a regular algebraic local rin.q of L which contain.s Rl such that

Ro

R

-+

S

--+

c V·

T

S· is a diagram .satisfying the conclusions of Theorem 9.1, then: 1. There is an isomorphism,

zn / Atzn

(b1 , ... , bn ) 1-4 b1v*(yd

D::

f* If given by

+ ... + bnJ/"(Yn).

2. f+ Ir acts on SO!JK(S)!( and the invariant ring is ~

A

-

7

R0K(R)K

('oJ

(l9@K(S)Kl*;r.

3. The reduced ramification index is

e = If·jrl= IDet(A)I· 4. The. '('dative. de.gr·e.e. 'is

f

= [K(V*) : K(V)]

=

[K(8) : K(R)].

It is shown in [35] and [32] that we can realize the valuation rings V and V* as directed unions of local rings of the form of Rand 8 in the above

9. RamificatiUIl of ValuatJoll.~ and Simultaneous Resolution

161

theorem, and the union of the valuation rings is a Galois extension with Galois group f* If We now give an overview of the proof of Theorem 9.2, referring to [35] for details. 11* is our valuation of L* with valuation ring V*, and v = 1/* I L is the restiction of v· to K. In the stateIIJent or the theorem f* is the value group of V· and r is the value group of il. By assumption, we ha.ve regular parameters (Xl, ... ,xn ) in Ro and (Yl, ... ,Yn) in S which satisfy the equations 'It

Xi

=5i

II

at' .1

Y.i

for 1 ~ i ~ n

j-1

of (2.8). We have the relat.ions n

V(:l:i)

= I:aijV*(yj)

E

f

j=l

for 1

< 'i < n. Thus there is a group homomorphism

zn /AtZn ~ f* /r defined hy 1. To show that this homomorphism is an isomorphism we need an extension of loeal mOllorniali:t.ation (proven in Theorem 4.9 [35]). 3 is

ilnmcdiate from 1 R{J -4 8 is monomial, and 11.0 ~ R is a weighted blow-up. Lemma 4.4 of [35] shows that if M* is the quotient field of S0K(S)K and A-l is the quotient field or R0K(R)K, then lvf*/M is Galois with Galois group Gal (1\!f* 1M ) Z7l/Azn f'V znjAtzn. Furthermore, (S~T«8)K)Zll/A'J'}1 ~ R0J({R)K. -

...

f'V

We sec that R 0K(R) !( is a quotient singularity, by a group whose invariant fa,etors are rleterlnined by f* If. This observation leads to the counterexample to global wea.k simult.aneous resolution stated earlier in this section. This example is written up in detail in [301. We will give a very brief ouLlille of the construction. With the notation of (9.6), if a valuation II of !((Y) has at least two distinct extemlions 111 and vi to K(X~) which have non-isomorphic quotients rifr and r~/r, then this pre~entl:l all obst.ruction to the existellce of a map Xl -} Y1 (with the notation of (9.6)) which is finite, and Xl is non-singular. If there were such a map, we would have points IJl and P2 on X}, whose local rings RI and R 2 are regular a.nd are dominated by 1/1 and 112 respect.ively~ and a point p on Yl whose local ring R is dominated by p. If we choose our valuatioTJ II ca.refully, we can ensure that R is a quotient of R 1 by fi/f and R is also a quotient of R'}. by f;/r. We can in fact choo8e the extended valntions so that we can conclude that not only are these quotient groups not isomorphic, but the invariant rings are not isomorphic, which is a contradiction. A

,..

,..,

AI

162

9. Rammcation of Valuations and Simultaneous Resolution

An interesting question is whet.her U,e condusions of local monomialization (Theorem 2.18) hold if the ground field ]( has positive r:haraet.eristk. Th j s is certainly true if trdegK (L.. ) = 1. Now suppose that trdeg[«(T/) - 2 and !{ is algebraically closed of positive charact.eristic:. Strong monomialization (a refinement. in TheoreIn 4.8 r351 of Theorem 2.18) is true if the value gronp r* of V* is a finitely generated group (Theoreln 7.3 [35]) or jf V* IV is defectless (Theorelll 7.:~5 [35]). The defect. is au invariant which is trivial in charadel'ist,ic z;ero, but is a power of p in an extension of chal'actF.rist,ie p flUlction fields. Theorem 7.;i8 [35] gives an example showing that strong monOluializ;ation rails if tnlegT«(L*) = 2 and char(K) > O. Of COIII'se the value group is not finitely generated, and V* I V ha.1:ol a. defect. This example does satisfy the less restrictive conelu:sions of local monomialization (Theort~m 2.18). It follows from strong monOlnializat.ioll j n J:hara,eteristic zero (a refinement of Theorem 2.1H) t.hat. local:sirnultaneous resolution is true (fof arbitrary trdegg(L*}) if !{ has characteristic zero and rational rank lJ* = 1. We cOIlsider this condition when trdegl«(l.. *) =- 2 and K is algebraically closed of positive characteristic. In Theorem 7.33 [351 it is shown that in many easel; lor:a] simulta.neous resolution does hold iftrdegJ((L*) - 2', char(!{) > 0 and the value group is not finitely generated. For instance, local simultaneous resolution holds if r is not. p-divisible. The example of Theorem 7.~8 [351 discussed above does in fact satisfy local simultaneous resolution, although it does not satisfy strong monomialization.

Appendix. Smoothness and Non-singularity II

In this appendix, we prove t.he theorems 011 the singular locus stat.ed in Scctioll 2.2, and prove theorems on upper Remi-continuity of order needed in our proofs of resolution. The proofs in Section A.I a.re based on Zariski's original proofS in [85].

A.I. Proofs of the basic theorems Suppose tha.t. P E A1( = spee(T
is den II eLl by

n}.n

= 0 A" d:l: 1

'TrLp

C

EB ... ffi 0 An cLa:n

au d(u) =. dXl + ... + ;:-d:Dn · aXl a:l.: 811.

11

d induces a linear t.ransformation of K (P)-vector spaces dp : 'rnp/(mp?

---i

nln ,p Q9 ]((P).

Define D( P) to be the image of dp. Since f.Jp",lI.P is a regular local ring, dim 1{(p) rnplm~ = n-r. Thus dinlK(P) D(P) < n-r, and dimK(p) D(P) = n - r if and only if dp : 'ml'/(mp)2 ~ D(P) is a K(P}-linear isomorphism. Theoreul A.I. Suppo.'U': Uwt P E AI( = spee(J<[xl, ... , x n ]) i.lJ a closed po'int. Then dilllK(p) D(P) .=- n if and only if J«P} is /u~l)a'f"(.Lble over K.

163

164

Appendix. Smoothness and Non-singularity II

Proof. Suppose that P E spec(K[:r.I, ... , xnD is a closed point. Let m,p C OAI'l,P be the ideal of P. Let (\!i be the image of Xi in K(P) for 1
(/1 (Xl ),1'2(:1:1, X2), ... , fn(xl, ... , x n ))

C (JAn ,po

By consinldion I c mp and K[Xl,.'" X n ]/ I ~ K(P), so we have I = Ulp. Thus !l(xt}, f2(Xll X2), ... , !n(Xl, ,Xn ) is a regular system of parameters in (JA'i.."P, and the images of h (Xl), h(Xl, :r.2), ... , fn(Xl' ... ,xn ) in mp/m,~ font! Ii. K(P)-basis. Thus dimK(p) D(P) = n if and only if rlp(/l) , ... ,dp(fn) are linearly independent over K(P). This holds if and only if J(f; x) har; rank n at P, which in turn holds if ann only if the determinant o ••

n

ali

I.J (f; ;1:) I = II ax' . 1 t

't=

is not zero in K(P). This condition holdfi if and only if !{(nl,'''' Cl:i) is separable over ]((0:1, ... ~Q'i-d for 1 < i < n, which is true if and only if J((P) is separahle over I<. D

Corollary A.2. Suppose that P E A"k =- spec(K[xl' ... ,xnJ) is a clo..,ed point such that K (P) 'io" 8f:])(l'f'Uble over K. Let mP c 0 An, p bf: thf: ideal of P, and suppmu~ Uw,t '/.tIt .. 0' UrI E mp. Then Ul, ••. , Un i..'l a regular system of lJannneter's in OAK,II if and only iJ J('ll.; ~r.;) has rank n at P. Proof. This fol1ow~ since lip : mjJ/m} only ifK (P) I:; separable over K.

--t

D(P) i" an iSOIlIorphism if and D

Lemma A.3. SUPPO,9C that A is a regular local ring of dimcn.,,';'on f) with maximal ideahn, a/fUlp C A is an equidimen,9'ional idenl such thatdim(A/p) = a. Then OJ

2

dim Aim (1' + rn"'/m ) < b - a,

and dhllA/m(l' + 'm.'l./m 2 ) = b - a if and only 'if Alp

7:8

a regular local ring.

Proof. Let A' = Alp, m' = mA'. k = Aim, A' /m,'. There is a short exact sequence of k-vector spaceR (A.I) 0 --t p/pnm,2 IJ+rn 2/m 2 -+ mlm 2 --t m' /(nl-,)2 = m.j(p+rn2) -+ O. 'V

f"v

The lemma now follows, since dimk m' /(m')'l. > a and A' I:; regular if and only if din1k 7T~' /(m.') '2 = a (Corollary 11.5 [13}).

0

A.1. Proofs of the ha...,ic tlleOl'emS

165

WP. now give two related results, Lemma A.4 and Corollary A.5.

Lemma A.4. Suppose that A is a regular local ring of dirnens'ion n with ma.T.'imal ideal m, peA is a prime ideal such that A/pis regular, and dirn(A/p) = n -".. Then /.heTf:~ e:l":i,'iit. reg'/J.la1~ parameters tLl, .•• ,Un in A such that p = (Ul, ... , u r ) and Ur+l, .. . , U'n map to regular parameters in A/p. Proof. Consider the exact. sequence (A.I). Since A and A' are regular, there exist rep;ular parameters Ul: ... ,Un in A such that ur+l,· .. ,Un Hlap to regular parameters ill A', and u 1 , ... ,ur are in p. (U1, ... , ur ) is thus a prime ideal of height l' cuntained in p. ThlIS P = ('lll,' •. , tL r ). 0 Corollary A.5. Suppose that Y is a subvariety of dimension t of a variety X of dimension n, 8uppmw. that q E Y 'is a non-singular closed point of both Y and X. Then there exist regular parameter's 'Ul, ... ,'UoT/ 'in (')X,CJ .'1'/1.r.h that I Y,q = (tL1 , ••• , Un-t), and Un-t+ 1, ... , Un map to regular parameters in Oy;q. If Vn-t-f-l' ... ,'Un u're r'egullJ,r pf1..mmf'.t,.~rs in (')Y,q. there exist regular parameters u 1, ••. , Un as above such that Ui maps to Vi for n - t + 1 < i < n.

Lemma A.6. Let I = (il, ... , fm.) CK[~1:1,"" x n ] be an ideal. Suppose that P E V(I) and J(f; x) has rank n at P. The.n K(P) is se.pu1'lLble lJ,l9t:~­ braic over K, and n of the polynomials f1' ... , f m form a system of regular parameters in 0 A.I, ,P' Proof. Letm be the ideal of P in K[Xl, ... , Xn]. After possibly reindexing the Ii, we may assume that IJ(/1, ... , In; :r.)1 ¢ m" By the Nullstellensatz, m is the intersection of all maximal ideals n of K[Xl,' .. ,xnl eontaining 'TT/" Thus there exists a closed point Q E A,( with maximal ideal 11, containing m such that IJ(iI, ... I fn; x) I ~ n, so that J(il," . I in; :1:) ha..~ rank 11, at n. We thus have dimK(Q) D(Q) = 11" so that K(Q) is separable over K and (/1, ... ,in) is a l'egular system of parameters in OAn,Q by Theorem A.I and Corollary A.2. Since I C rti C it, we have P = Q. 0

Lemma A.7. S'lJ.pposcthat PEAK = spCC(K[Xl,' .. , In]) with ideal m C K[X1, ... , In]

is such thut m. hus he'ighl, n - rand 1.1l ••• ,t n - r f':: mm C ()An ,P • Let Qi be the image of Xi in J«P) for 1 < i < r. Then the dete17ninf1:nl I./(tl, ... , t n - r ; x r +},···, xn)1 is non-zero at P if and only if 01, ... , Or is a separating transcendence blLSis of K(P) ove7' K and t 1, ... ,tn.-r i,1) a regular system of parameters in 0 A" ,p. Proof. Suppose that IJ(tl, ,tn -r; :Z:"'+b ••. ,xn ) 1 is non-zero at P. There exist So, 81, ... , 8 n- r E K[X1, ,InJ such that 8i E I<[xI, ... ,xnl fur all i, .~o ¢ Tn and ~ = ti for 1 < i < n- r. Then

1.I(sl,'.' ,Sn.-r; Xr+l,· •. ,xn)1

Appendix. Smoothness and Non-singularit.y II

166

is non-zero at. P. Let K* = ]«(a1,

1

K* [X r + 1, and let

sf for 1 < i < n -

a r ), let n be the kernel of

,Xn ]

~K (P),

r be the image of Si under

K[Xl, ... ,:.L:n l ~ [(*rx r

11,· .. ,

x n ].

n is the ideal of a point Q E A/f<"r. We have K(P) = K*(Q), and at

q,

IJ(s'1,·· ., s,;~_,/.; Xr +l,··· ,xn }\ =F O.

(sr, ... , S'~~-1') = ](* [X r +l, ... , xn]rr. Since

By Lelllina A.f), K*(q) is separable algebra.ie over is a regular system of parameters in

('J .. n-,- Q noA·...

,

}{'II anu

trdegK K(P) = r, O!l, . . . , O!r is a separating transcendence basis of K(P} over ](. Thus K* c K[X1, ,Xn]m and ]([Xl,

,xnJm - K'" [X r +1, ... ,xn]n,

so that tl, .. . , In-r is a. regular systeIIl of' parameters in O/9/i<,P

= K[Xl, . .. ,:l:ll.].,-,t.

Now suppose that 01, ... , Or is a separating transcendence basis of K (P) over K and tl, ... , t n --l • is a l'egu Iar system of parameters in 0 A"k ,P ]([XI, ... ,XnJm. Let ](* - K (01, , D: r ), and let n be the kernel of ,~l:nl --+ K(P).

K*[:l:1'+1,

n is the ideal of a point Q E

A~-:·/.

Then K*(q) = K(P) and K"'(q) is

separable over K*. We have a natural inclusion K* C [([Xl, ... ,Xn]m. Thus K[:r.l;···, :l:,/Jrri

=

K*[:L: r

lb··

0' xonl on ,

and tI,' .. , t n - r is a regular system of parameters in K* [X 1'+ 1, ... ,X11.].ii.. By Corollary A.2, 1.1(tl:"" l. n- 1·; :l;,/,tb"': xn)1 i= 0 at Q, and thus it is also i= 0 at P. 0

Theorem A.B. 8'l/,lJpOSe that P E A1< = spec(K[xl, ... l xnJ) and tt l ' •• , tn-to are regular parameters in ('JAn ,p. Then rank(J(t; x)) = n- r at P if and only if K(P) is separably generntp,d 01'''~1' K. Proof. Let Qi, 1 ~i < n, be the images of ideal of P in K[X1, ... , x n ].

Xi

in [( (P), and let m be the

Suppo~e

that rank(J(t; x)) = n - r at P. After possibly reindexing the Xi and tj, we have IJ(tt, ... , t n - r ; X r +l, ... ,:1:11.) I =1= () at P. Now K (P) is separably generated over I< by Lemma A.7. Suppose that K (P) is separably generated over K. Let (1, ... , (r be a separating transcendence basis of K(P) over K. There exists \}lj(:C) E [([Xl, ...

,xn ] for () < j < r with \II 0 (:t)

rt

m. such that (j

='

:~~~~. Let

A.l. Proofs of the b81:>ic tlll:~ul'em.':i

167

n be the kernel of the K-R.lgehra honlOffiorphism K[Xt, . .. , xn+r] ~ J«P) defined by Xi ~ Cti for 1 < i < n, Xi 1-+ (i- Tl if n < ,i. n is t.he ideal of a point. Q E A,/]/r. Let <1'O(X),
= <1'i(X)

ti

(f>o (:1:)

for 1 < i

< n - r.

Let. n-1'+,i(x) = xn+jWo(x) - Wj(x) for 1 < .j < r. Let I C K[X1 1·· ., X n ,7'] be the ideal generated by 1, ... , ni thus 1 C n. We will show that ~1, ... , ~n are in fact a regular system of parameters in

V I"OJ( "nl1' Q , II. suffices to show that if

= K[xt, ...

I

xn+r]n.

F(x) E n, then there exists

A(:c) E K[:l:j: ... , a: n +r ]-· n such that AF E I. Given such an F, we have an expaUl:iiOIl r

'lTo(x)S F(x) =

E Bj(x)tPn-r-u(x) + G(~l:l,

:C2,· .• , :1:1/.),

j=1

where Hj (:?:) E: K[Xl,' .. ,Xn,+r] and s is a non-negative integer. Now F E n implies GEm. Thus then! existsa.n expansion Tl

G

l'

= Lfiti i=l

with Ii E Kr:CI, ... I :i:nl'///" and t.here exist h C K[Xl, ... , x n ] Ci E J([XI, ... : x n ] such that

-

{m} and

7'

hG -

L Ci4'i(X) i.,-!

and It \Il o(:L: ) ~ P (:I;) E J. Since q,1, ... ,CPn is a regular system of paraHleterl:! in K[:I:;J, ... , x n +1']n. and (1, ... , (r is a separating transcendence basis of J«(Q) over K, it follows that

IJ( 1, ... '
Proof that smoothness at a point is well defined Suppose that X is a variety of dimension s over a field K and P E X. In Definition 2.6 we gave a definition for smoothness of X over K at P which

168

Appt!Ildix. Smoothness and NOll-singIlJa.l'it.y II

depended on choices of an affine neighborhood U -= spec(R) of P and a preAentation R ~ K[Xl' ... ,xnl/I with I = (iI, ... ,1m)' We now show that this definition is well defined, of all choices U, I and ;z;.

80

that it is independent

Consider the sheaf of differentials n1/ x. We ha.ve, by the "second fundamental exact sequence" (Theorern 25.2 [66]), a presentation

(A ..2)

(K[ X!J

... ,X n ])

1f/.

J(f;x) (')1 ~ HKlxl, ... ,xn}/K®

Tensoring (A.2) with K(P), we see that only if J(f; x) has rank n - s at P.

R

n I

0

--tHRI1<.--t·

n1/K ® K(P)

h~ rank 8 if and

Proof of Theorem 2.8 Let s = dim X. There exist.s an alline neighborhood U -= spec(R) of P in X such that R - .KrXl,'" ,xnjjI, I = (ft, ... ,fm)' Let m.p E sper,(K[:r:l" ", x n ]) be the ideal of P in A'k. Let

t = dim(K[xl,'" ,xn]/mp). Suppose that J«P) is separably generated over K etnel P is a nonsingular point of X. Dy Theorem A.S, rip : ntP/Tn~ --t D(P) is an isomorphism. By LemmaA.:~, dim}<.(JJ) dp(I) = n-s (take A = OAn,p, P = I.",p ill the Atatement of' the lemma, so that b = n - t, a = s- t), which is equivalent to the condition that J (/; x) has rank n - }:4 at P. Thus X is smooth over

Kat P. Suppose that X is smooth at P, so that .1(/; :1:) has rank n - s at P. Then dimT((p) dp(I) = n - s, so !.hat diIn[«(p) (I + m~)/m~ > n- s. Thml P is a non-singular point of X by Lemma A.3.

Remark A.9. With the notation of Theorem 2.8 and it'R proof, for any point P E V(1), we have · 2 2 dllnx(lJ) (I + mp)/mp < n - s

hy Lemma A.3, so that J(f; x) has rank

< 11. -

8

at P.

Proof of Theorem 2.7 It suffices to prove that the locus of smooth points of X lying

an open affine subset U = spec(R) of X of the fbrrn of Definition 2.6 is open. Let A = In-s(J(fj x)). Then P E U - V(A) if and only if J(f;x) has rank 011

169

A.2. Non-singularity and uniforIIll:£illg jJal"FJJl1ctcrs

greater than or equal to 11. - 8 at P, which in turn holds if and only if J (J; :1:) ha:s ra.nk 11. - s at P by Remark A.9.

Proof of Theorenl 2.10 when K is perfect SlIppose that Ie is perfect. The openneS8 of the set of non-singular points of X follows from Theorem 2.7 and Corollary 2.9. Let 1} E X be the generic point. of an irreducible component of X. Then CJX,1} is a field which is a regular local ring. Thus 71 is a non-singular point. We conclude that the non-singular points of X are dense in )(.

A.2. Non-singularity and llniformizing parameters Theorem A.lO. SnPlw,f,c that X is a variety of dimension 'f' (mer a field K, and P E Xis a closed point s'i.u:h that ~'<. is non-"singular at P and 1« P) is se.pa1uble O1Jer K. 1. Suppose that Yb ... ,y.,. are n:gular parameters in 0 X,p. The.n

n~/l(,p

=.

dy10X,p E9'" Ee cly.,.Ox,p.

2. Let 'in be f,ht:~ idf'ALl of P in OX,p. Suppose that It, .. ., j~. E 7n C OX ,P and dh, ... ,d!1' generate n!xl K,P as an OX,? module. Then .f1, ... ,fr are 1'egular Immmetcrs in OX,p.

Proof. V,le can restrict to an affine neighborhood

or X, and assume that

X = spec(I([xl' ... ,xn ]/ I). The conclilliioJl8 01' part. 1 of the theorem follow from Theorem A.S when

X

_An

-

fi,(,

In the general case of 1, by Corollary A.5 there exist. regular parameters (YI"'" y.,,) in A = K[:I:l,.'" xnh=n (where m is the ideal of P in spec(K[a:t, ... ,In]) such that fA = (Yr+l"'" Un) and (Yl"" ,Yr ) map to the regular para.llleters (Yl, ... , Yr) in B = AI I A. By t.he second fundamental exact sequence (Theorem 25.2 [66]) we have an exact sequeuee 01' B-modnJes

IAII 2 A --+ n~/K ®A B

---t

n1I K'-) o.

This sequence is thus a split short exact sequence of free B-modules, with n~/l< ~ dYtBm·· . ffidYr B , since I A/ {l. A ~ Yr+l B (;B ..•
Appendix. Swoothness and Non-singularity II

170

n~/](,p by part 1 of this theorem. There are

a'l)

E

R such that

l'

Ii

=

L aijYj,

< r.

1 5: j

j=l

We have expressions

dfi -

'I'

r

j-l

.j=l

L ai.jdYJ (2: daijYj)'

Let 'Wi = E~=l u-ijdyj for 1 < i < r. By Nakayama's Lemma we have that the 'W't generate n\/K.p as an R-illodule. Thus l(eLij)1 .fJ. ill and (aij) is invertible over R, so that h, ... , fr generat.e 'In. 0

Lemma A .11. Suppose that X is a variety of dimcn..,'ion '1' O'/lf;'" a peTfect field !{. Suppose that P E X is Q. dmied point. s'ueh that X is non-singular at P, and 'Y1 , ... : Yr (L1't: Toegular parameters in Ox IP, Then there er,'l.'"its ([,'11. anint:: u,t':ighborhood U = spec(R) of P in X such thal Yb ".. ,y." E R, the natural inclusion S = K[Yl, ... ,Yl'] --+ H induces a morphismrr : U --+ A K such that (A.3)

nkj T< = dYI H (f) •

0

0

EV tlY'rR,

a

Derj( (R, R) - IIomn(nk/K' R) = 1)0

Yl

REB"" EB

a

-8 R Yr

(with {j~~ (clYj) ={Jij), and the following condition hold.,;. 8U1JPO.'ojt; tha.t ll, E U 'i.'l (J. clo~"tell point, b = 1r (a), and (Xl, ...• X r ) arc n:g1l.lo:1' ptL1'([,mde'T's in (')A 1\: ,b' Then (x 1, ... , X r) me:. n;guLtL'1' Jlarumeters in (')x ,a· 'JI1, ... ,Yr are called unifonnizing para.meters on U.

Proof. By T'heorem A.1 0, there exists an affine neighborhood [} = spec( R) of P in X such that Yl, ... , 'JIr E Rand dYl, . .. ,dy,. is a fJ'ee basis of 11k/i<. Consider the natural inclusion .5' ."",. KrYl,"" Yr] -+ R, with induced morphism 1r : sp~c( R) ---+ spec(S). Suppose that a E U is a closed point. with maximal ideal m in R. and b = 1r(a)~ with maxima.l ideal n in S. Suppose that (Xl •... , Ir) flJ"e regular pa.rameters in B = Sn. Let A = Rm;. By COllst.ruction, U1/J< i~ generated by dyl, . .. , dYr a..<; an 8-module. Thus n1/K is genera.ted by nh/T{ as an A-module. By 1 of Theorem A.10, d:I:l l • .• ,dxr generate Uh//( as a B-module, and hence they generate n~/J( as an Amodule. By 2 of Theorem A.l0, Xl, ... ,Xr is a regular syst.em of parameters in A. 0

171

A .3. Higher derivations

A.3. Iligher derivations Suppose that !( is a field and A is a K -a.lgebra. A higher derivation of A over K of length m is a sequence

•

if m

< oc, and a sequence D = (Du, D il

if 'Tn

-

00,

. .. )

or K-lillear' maps Di : A ~ A such that Do =

id and

'/,

(AA)

Di(XY) =

L

Dj(x)Dj-i(Y)

j-l 1'01'

1 <;i

< m, and x, yEA.

Let t be an indetenninate. f) - (Do, D 1 , ••• ) is a higher derivatign of A over K (of length 00) precisely when the map E t : A ~ A [[1,]] defined by

Et(f)

= 2:ro f)i(!)fi isa K-algcbra homomorphism with Du(f) =

j', and

D = (Do, D 1 , ... 1 D m ) is a higher derivation of A over !( of length m precisely whcn the map E t : A -+ AltJl(tm +1 ) defined by Et(f) - L~ 0 D'i(f)ti is a K-algebra homolllOrphi<3m with Do(f) = .t.

EX81nple A.12. Supposc that A = 1([Y1, ... ,1IrJ is a polynOlIlial ring OVfH' a field 1(. For 1 < i < 'f', let f j be t.he vector in ~P" with a 1 in the .ith coefficient and 0 everywhere el1=le. .For f = 2:: ail,... ,i"yi t ••• y~r E A and mEN, define

* (f). D mej Then 00

.;;;;;:..

~ L ail!, .. ,i

n; = (Do, D: D2tj ,...) i~

for 1

< j < r.

j

,

j

m ) Ylil ... Yjij-m ... Yri 7, .

12'; are itcrative;

Die) (') DkfJ = 1'01'

(i

higher derivat.ion of A over ]( of length

fL

Observe t.hat. the

r

i+k) (i

that is,

D(i+k)!J

all i, A:. Define

for ('il ~ ... , i 1·) c; N'·. The

Dr ··,·r.; commute with each other. &<1~'

The DtJ.,... ,lr are called H8S8e-Sclmlidt derivat.ions [56]. If K has characteristic zero, then

Dt-t,. .. ,or .; --

1

'a Yl'"

I,; I ••. '; "1'''2· "r·

,;

a'i, +···+i,.

--<-:-.------:-, • tl

!l t 7 , UY'f'

Appendix. Smoothness and Non-singularity II

172

Lenuna A.l3. Suppose that X is a variety of dimension r O'lJer a perfect field K. Suppa.se that P E X is a closed point .rI'ur.h that X is non-singular at P, Y1, ... , Yr are regular paramete'l's in () X,P, and U = spec( R) is an affine neighborhood of P in X s'Uch that the conclusions of Lermna A.11 hold. Then there (L1'e. diffe1'ential operators Di1 •... ,ir on R, which are uniquely detennined on R by the differential openLt01's n;] ,... ,i r on S of Example A. J2, such that if f t Ox, p and

}"- L a'

l . y iT . "'1 71· i••• r

11, ... ,t r

-

in Ox,p "-i K(P)[[yl, ... ,Yr]], where we identiftJ K(P) with the coefficient field of Ox,P containing 1<, then ail, ,ir = Di], ... ,ir(f)(P) is the residue of D il ,...,i r (f) in K(P) jor all indir.e.fI it, , ir' Proof. If.K hal'3 characteristic zero, we can take

(A.5)

D·

ail +··+'i.,.

1

, - - - _.."-,.--------,!-l. i 1 fJ .i . ~1 • t.,,..I VYI ... Yrr

tl,·· .,1 T

-

•

'...:

Suppose that K hafl characteristic p ~ O. K-algcbra.-, S = K[Yl , ... ,'Y1']

-+

Consider the indu~ion of

R

of Lelnma A.11, with induced morphism 7r :

U = spec(R)

-+

V

= spec(S).

Let Dj for 1 <j < r be the higher order derivat ion of S over K defined in Example A.12. We have = 0 by 1 of Theorem A.10, (A.3) and the l'first flUldamental exact sequence" (Theorem 25.1 [66]). Thus 7r is non-ramified by Corollary IV.17.4.2 [451. By Theorem IV.18.10.1 [45] and Proposition IV.6.15.6 [45], 1r is etale.

nk/s

We will now prove that Dj extends uniquely to a higher derivation D j of Rover K for 1 < j < 7'. It suffices to prove that (Do, D:', ... , n:'u,j) extends to a higher derivation of Rover K for all m. The extension of Do = ill to Do = id is immediat.e. Suppose that an extension (Do, D 1 , .. . , Dm ) of (Do, j , ••• , D:'tt j) has been constnlcted. Then we have a commutative diagram of K-algebra homomorphisms

D:

R .!t Rlt]!(t m +1 )

i

s L where

1 R[t]/(trn+2 ) m

g(a)

= LDieoi(a)tj, i=O

A.3. Higller derivations

173

m+l

f(a) =

L

Di€j (a)t j .

i=O

Since (t lH +1 )R[t]/(tm +2 ) is an ideal of square zero, and S ~ R is formally etale (Definitions IV.17.3.1 and 17.1.1 [45]), there exists a unique K-algebra

homomorphism h making a commutative diagram R ~ R[t]/(trn.+l) It.

i '\. 8

i

.L

R[t]/(t rnt2 )

Thus there is a. unique extension uf (Do I D:j derivation of R. Set D·1.1, •.• ,7.,.. -- D·'lrf:'1" 0 . . . 0 D·1.1f"1 • ..

, .••

,D,m+l)€j) tu a higher

Let A = Ox,p. Since K(P) is separable over K, and by Exercise 3.9 of Section 3.2, we can identify the residue field J((P) of A with the coefficient field of A which contains K. Thus we have natural inclusions

K[Yl,"" Yr] cAe

A

"01

K(P)[[Yl,'" ,Yr]]' h

We will now show that D f,j extends uniquely to a higher derivation of A over K for 1 < j < 7'. Let 111, be the maximal ideal of A. Let f. be an '" indeterminate. Then AHt]] is a complete local ring with maximal ideal

n = m,A[[t]] + tA[[t.]]. Define Et : A ~ A[[t]] by EtCf) = L~ 0 Dmj(f)fn. E t is a K-algebra hOJIlOIIlOrphism, since D F':J is a higher derivatiull. Et is continuuus in the m,-adic topology, as Et(m,U) C n U for all (1,. Thus E l extends uniquely tu a K-algebra homomorphsim A--) A[[t]]. Thus n~:J extends ulliquely to fL

.. higher deriV'd.tiun of A over K.

h

.

We can thus extend Dil,. .. ,ir uniquely to A. Since K(P) is algebraic over " K, and DiI,...,i,. extends Di1t...,i,., it follows that Dil,... ,i,. acts on A as stated in t.he t.heorem. 0

Remark A.14. Suppose that the assumptions of Lemma A.13 hold, and that K is an algebraically closed field. Then the following properties hold. 1. Suppose that Q E U is a dosed puint and a is the corresponding maximal ideal in R. Then a = (Y1' ... , Yr ), where Yi = Yi - Qi with (Yi = Yi(Q) E K for 1
f =L

ait ,... ,iryl1

•••

y!,:r

174

Appendix. Smoothness lwd NUll-singularity II

in K[[Yll'.' l Yr]] = Retl where lL·il, ... ,ir = Di],....ir(f)(Q) is the re:sidue of DiI,... ,ir(f) in K(Q) for all indices il,.'" i 1' .

Proof. The ideal of rr(Q) in S is (Yl - Qll ... ,Yr - ar) with ai = 'J.1i( Q) E K. Let Y'i = Yi - ai for 1 < i < r. We have a = (Yl' ... l fir) by Theorem A.tO 2, and 1 follows. The derival:ionR ~l"'" 8~r are exactly the derivations

.jL, am ... ,.,.fL . 011t· on S of Example A.12, so they extend to the differential opp'l'at.or·sOi1 , ... ,i r Ull /3 uf Example A.12. Now 2 follows from Lemma A.l:i applied to Q and Yl' ... , Yr' 0

Lemma A.15. fJf'.l no/,nt:ion be us in Lemmas A .11 and A .13. In particular, we have. a fixed closed point P E U -= spec(R) and differential operat01'8 Dii,... ,i,. on R. Let I c R be an ideal. J)f~Jint~ an 'ittenl Jm,u C R by

.Jm.,u = {f)'it ,... ,ir (/) I J E lund i 1 + ... + i r < m}. SlllJ]JO.':U::

Uw.t Q E U is a closed point, with corresponding ideal mQ in R,

and suppose that

(Zll ... , zr)

is a regular system of p(J,1'(J,m.d.t~'r8 'i.n

'rna.

Let.1J i1 ,... ,ir be. the. differential operators constructed on OW,Q by the construction of Lemmas A.ll and A.18. Then (.Jrn.,u )rn.Q

= {Dil,... ,·ir (f) I JET

and il

+ ... + i r < mo}.

Proof. When!( has characteristic zero\ thi., follows dirp,dly rl'OIll differentiation and (A.5). Assume that K has positive characteristic. By Lemma A.13 and Theo.:.' rem IV.16.11.2 [45]\ U is differentiahly smooth over spee(I{) and {Di1, ... ,'i r I il, ... 1';'1' < 'm} is an R-basis of the differential operators of order less than

m on R. Since {dz 1 ,.·. dz r } is a basis of l

Ok

m'Q

/K

(by Lemma A.11),

{D i1 ,... ,ir I it, ... 1 i r < m} is an RmQ-basis of the differential operators of order less than 1n on Rm Q by Theorem IV.16.11.2 [45]. The lemrna follows.

0

A.4. Upper semi-continuity of Ilq(I) Lemma A.16. Suppose that W i.Ii a non-sing'IJ,lar' 'L1l1;l'ie.ty o'ue.r' a perfect }I.dd K and K'is an alge.braic field extension of !(. Let ",r' = W Xl( I(', with projection 7rl : W' ---+ W. Then VV' is a non-8ing'ILla/r 'Im,1"i.el.y UVf:T K'. Suppose that q E W i.'i (/, dO::led puint. Then there exists a closed point q' E W' s'llch that rrdq') = q. Furthermore,

1. Suppose that (II, ... , fr) is a regular system of parameters in eJtv.q' Then (ft, ... , f1') is a regular .9Y.9tf~rn of pa1'am,e/,f1'S 'in Ow' ,q' .

A.4. Upper semi-colltinuity of lIq (I)

175

'----'--'----'-----------------

2. If I

c

OW,q is un ideo,l., l!u;n

(IO\1.'1,q') rl On",q

= E.

Proof. The fact that. 'W" is non-Flingular follows from Theorem 2.8 and the definition of smoothness. Let q' E 11'7' be a dosed point Ruch that 71"1 (q') = q. We first pmve 1. Suppose that U = spee(R) is an affine neighborhood of q in W. There is a presentation R. = K [Xll ••• ,xnl/I. We further have that 7r 11 (U) = spec(R'), with R' = [<'[Xl;. " 1 xnll I(.K'[:1:1," ., :1: 71.]). Let. m be the ideal or '} in K [Xl, ... l x n ]! and n the ideal of q' in [(' [XI, ... , x n ]. By the exact sequence of Lemma A.3, I. here exists a regular system of paralneters f l' ... , f n in ]([Xll ... , xnJm, l:iueh that Ini = (lr+l"'" f n)111 and Ii maps to Ii for 1 < i < r. By Corollary A.2, the determinant (J(ll)' .. , In; :1:1, ... ,Xn ) I is non-zero at q, so it is non-zero at q'. By Corullary A.2, 111"" in is a regular system of parmneters in K'[xlt ... , xnJn. Thus /1" .. , f r is a regular system of parameters in R~, - OW'.q'· We will nuw prove 2. Since n Inaps to a maximal ideal of OW,q "

,

Q<)K ](',

. . I .-

OW',q' =- (OW,q ®K K )n

c,j. denotes cOlnpletion with respect to nOw,q ®K K').

A

Thws ('Jw',q is faithrully lIat over 0"",(1' and (IOw',q' )nOW,q - I (by Theorem 7.5 (ii) [66]). n

Suppose that (R, m) is a regular local dng r.ontaining a field K of characteri8tic zero. Let K' be the residue field of R. Suppu~e that J c R. is an ideal. We define the order' or J in R to be

lJR(J) = max{b I J C w,h}. If J il:i a locally principal ideal, then the order of J is its multiplicity. However, order and multiplicity are different in genera1. Definition A.17. Suppose that q is a point is an ideal sheaf. We denote Ilq(J)

If X

c 11'7

i~

011

a variety

llT

and J

c Ow

= IJOw.q(JOW,q).

a subvariety, we denote

llq(X) = llq(Ix).

Remark A.IS. Let notation be as in Leulma A.16 (exeept. we allow K to he an arbitrary, not necessarily perfect field). Observe that if q E Wand q' E W' are such that 7r(q') = q and J c Ow is all ideal sheaf! then IIO W ,q (Jq)

= 11,;:,

v~Y.q

(Jq(-jWq) ,

= lIo,v' .11, (.JqOw' q') = VA'-'\it./ ,q' (JqOw' ,q')' r

1.

Appendix. Smootlmess and NOll-singularity 11

176

Suppose that X is a noetherian topological space and I is a totally ordered set. j': X ---+ I is said to bean upper selui-Gontinuous functiuIl if for any CI:: E I the set {q E X I f (q) > O'} is a closed subset of X. Suppose that X is a subvariety of a non-singular variety W. Define 1/X : ltV

---+

N

by 1/x(q) = 1/q (X) for q E W. The order 1/q (X) is defined in Definition A.17.

Theorem A.19. S'Uppo.se thai, K i."4 a perfed field, W is a non-singular' variety over 1< and I is an ideal sheaf on W. Then lJq (l')

: W -+ N

is an upper semi-continuous function. Proof. Suppose that mEN. Suppose that U = spec(R) is an open affine subset of W such that the conclusions of Lemma A.II hold on W. Set 1 = l'(lJ, I). With the notation of Lemmas A.II and A.13, define an ideal Jm,u C R by

Jm,U = {Dil....,ir(g) I gEl and il + .. ·i r < m}. By Lemma A.15, this definition is independent of the choice of Yl, ... , Yr satisfying the conclusions of Lemma A.II, so onr definition of .ImP determines a sheaf of ideals Jm on W.

We will show that V(Jm )

= {1] E W I zl1](I) > m},

from which upper semi-continuity of zlq(I) follows. Suppose that 'rJ EW' is a point. Let Y be the closure of {o7J} iII W, and suppose that P E Y is a closed point such that Y is non-singular at P. By Corollary A.5, Lemma A.II and Lemnla A.I5, there exists an affine neighborhood U = ~pec( R) of P in W such that there are Yl, ... ,Yr E R with the properties that Yl = . . . = Yt = 0 (with t < r) are local equations of Y in U and Yl, ... ,Yr sa.tisfy the conclusions of Lemma A.II. For 9 E Ip, consider the expansion .. yi 1 Y=~ L..J (J,'~ I •... 'tr 1

•••

'I/

T ;;IT

in OW,P ""' K(P)[[Yl, ... ,Yr]], where we have identified K(P) with the coefficient field of OW,P containing K and (with the notation of Lemma A.13) ail, ... ,i,.

= Dil,....i,r(g)(P).

177

A.4. Upper semi-continuity of vq(I)

Then 77 E V(Jm ) is equivalent to Dit,... ,i'r(g) E (Yl,"" Yt) whenever g E Ip and h + ... + i r < m., which is equivalent to =

uil, ... ,ir

if 9 E 'Lp and il

+ ... + it < 111"

Dill ... ,ir(g)(P)

=0

which in turn holds if and only if

,.

9E

IY:p n OW,f' = T~p

for all 9 E Ip, where the last equality is by Lemma. A.16. Localizing Ow.P at 77, we see that 1I77 (I) > m if and only if 7] E V(Jm ). 0

Definition A.20. Suppose that W i::; a non-singular variety over a perfect field K and X

cW

is a su bvariety.

For bEN, define Singb(X) = {q E W I vq(W) ~ b}. Singb(X) ilS a dosed subset of W by Theorem A.19. Let

r.= max{llx(q) I q E HI}. Suppose that q E Singr(X). A subvariety H of an affine ueighborhood U of q in W is called a hypersurface of maxllla.l conta.ct for X at q ir

1. Singr(X) n U c H, and 2. if

Wn

1r

. ~ ... ~

WI

IT l

-t

W

is a sequence of monoidal transforms such th.at for all i, 7ri is centered at a non-singular subvariety Yi C Singr(Xi ), where )(i is the strict transform of X all Wi, then the strict transform H n or H on Un ~ Wn Xw U is such that Singr("l[n) nUn c Hn . With the above . assumptions, a. non-singular codimellsion one subvariety H of U = spec(OW,q) is called a formal hypersurface of maximal contact for X at q if 1 and 2 above hold for U = spec( ('?w,q ).

Remark A.21. Suppose that W is a non-singular variety over a perfflct .. field K and X c W is a subvariety, q E W". Let T = VW,q, and define

Singb(Ox,q) = {P

E spec(T)

I v(Ix,qTp) > b}.

Then Singb (O.x,q) is a. dosed set. If J C Ow is a reduced idea.l sheaf whose support. is Singb(X), then JqT is a reduced ideal whose support i::l

Singb(Ox,q).

Proof. Let U = spec(R) be an affine neighborhood of q in W satisfying the cunelllsions of Lemma. A.13 (with p = q), and let notation be as ill Lemma A.13 (with]J = q). With the further notation of Theorem A.19 (and I = Ix) we see that r(U, J) = vi Jb,U. JqT is a reduced ideal by Theorem a.6.

178

Appendix. Smoothness and Non-singularity II

Suppose that P E Singh(OX,q). Then 1JTp (g) > b for all 9 E Ix,q. Thus g E p(b) C T~ and there exists h E T - P such that gh E ph. Now induction in t.he formula (AA) shows that Dit,... ,ir(gh) E pb-il,,,.,ir and Dh,... ,i,.(g) E p(b i,,···-ir ) C p if il + ... + i r < b. Thus JqT c P. Now suppose that P E spcc(T) and JqT C P. Let S = OW,q. If J q = Pln· . ·npm is a primary decOInposition, then all primes Pi are minimal since.Jq is redueed. By Theorem 3.6 there are primes Pij in T and a function u( i) such that PiT = ~1 n ... n PiO'(i) is a, minimal primary decomposition. Thus JqTpij = P;,Tpij = PijTpij for all i,j. We have nij Pij C P, so Rtj C P for some i.,j. Moreover, IX,q C PPSPI. impliel:! IX,q C Pi~Tplj' which in turn implies IX,q C pbTpij' Thus VTp(Ix,qTp) > band P E Singb(OX,q). 0

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Index

Abhyankar, 12, 28, 42, 45, 130, 131, 152, 153, 157, 158, 160 Abramovich, 9 Albanese, 55 approximate manifold, 106 AtiYM , viii

intersection reduced set-theoretic, 62 scheme-theoretic, 2, 62 i80latAd subgroup, 134

Bierstone, 61 nravo~ 61

Karu, 9 Kawasaki, 11 Kuhlmann, 1::S

Jung, 55

Cano, 13 Chp.va.lley, 24 Cohen, 22

completion of the square, 24 Cossart, 131 curve, 2 de Jong, 61, 131 derivat.ions

HM8e-Schmidt, 171

higher; 171 Eiscubud, viii EnciIl8."l, 61

c.xeeptiona! divisor, 2U, ::SY, 40 exceptional locus, 39, 40 Giraud, 70 Harris, viii Hartshorne, viii, 2 Hauser, 13, 61, 132 Heinzer, 13, 157 Hensel's Lemmu, 23 llironaka, 2,15,61,62,130

hypersurfacc, 2

Levi, 55 Lipman, 130 loca! uniformization, 12, 130, 13R, 147, 156

Macaulayfication, 11 Macdonald. v iii MacLan~, 137 Matsulci, n maximal cont,ar.t forrum curve, 2R formal hypp.rsnrfa.ce, 177 hypcrsurfa.cp., 56, 61, 70, 131, 177 Milman, 61 Moll,61 monoidnl t,ram;form, 16, 39, 4U monomioliza.t.ion, 13, 157 morphism biralional, 9 monomia.l, III propel', 9 multiplicity, 3, H~, 25, 175

Nagata, 11 Narasimhan, 131 Newton, 1,3, 6, 12

185

186

Newton Polygon, 34 llormali:tation, 3, 10, 12, 26 ord, 22, L38 oreier, 3, 19,25, 138 £I,,{q), 63 £IR, 17!i £IX, 63, 176 £Iq (.1), 17!i vq(X), 17:1 upper semi-continuOllfi, 17ft Piltant, 13 principalization of ideals, 40, 43,99 Puiseux,6 Puiseux series, 6, 11, 137 resolutiun of differential forms, 13 r~ulution uf indeterminacy, 41, IUU resolutiun of l>ingularitics, 9,40,41,61, IOU 102 curvc, 10, 12, 26, 27, 30,38, 39, 42 cmb(...udcd, 41, 56, 99 (~llLi vnrio,nt, 103 excellent surfacc, 130 positive characteristic, 10, 12, 13, 30, 42, 105, 152 fmrfacp., 12, 45, 105, 152 three-fold, 12, 1112 resolution of vector fields, 13 • nng affine, 2 regula.r, 7 Rotthaus, 9, 13

scheme Oohen-Macaulay, 11

excellent, 9 non-singular, 7

quasi-excellent, 9 smooth, 7 Schilling, 137 Scidenberg, 13 Serre , 10 Slmnnon, 151 singularity Sillg(J, b), 66 Sillgll(X), 45, 105, 177 Sing,,(0 x.II)' 177

SN(;s, 29, 40, 49 Spivakovsky, 13 surface, 2 tangent space, 8 Tcitosier, 13 three-fold, 2

Index

transform !>trict, 20, 25, 38, 65 total, 21, 25, 37

weak, 65 Tsehirnhauseu transformatiou, 24, 28, 51, 56,61, 67,70

uniformizing parameters, 170 upper semi-continuous, 176 valua.tion, 12, 133 eiimetlsion, 1M discrete, 134 rank, 134 rational rank, 136 ring, 134 variety, ~ integral, 2 quasi-complete, 15 !;ubvaricty, 2 Villwl1ayur, 61 Wa.lker, 55

Weierstrass preparation theorem, 24 Wiegand,13 Wlodarczyk, 9, 61 Za.rifiki, 1, 12,55, 133, 147, 148, 152, Hi6

Titles in This Series 63 Steven Dale Cutkosky, Resolution of sinp;ularities, 2004 62 'l'. W. Korner, A compa.nion 1,0 l.\llalysis: A second first and first second coursc in analysis, 2004 61 Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: Differential geometry via moving frames a.mI exterior differentia.\ systems, 2003 60 Alberto Candel and Lawrence Conlon, Foliations 11, 2003 59 Steven H. Weintraub, Representation theory of finite groups: algebra and arithmetic, 200:\ 58 Cedric VlIlanl, Topics in opt,imal transpurtation, 2003 57 Robert Plato, Cuncise numerical mathematics, 2003 56 E. B. Vlnberg, A course in algebra. 2003 55 C. Herbert ClcmeI1~, A scrapbook of complex curve theory, secund edition, 2003 54 Alexander Barvlnok, A course in convexity, 2002 53 Henryk Iwaniec, Spl'Ctrul methods of automorphic formll, 2002 52 Ilka Agricola and Thomas Friedrich, Glubal analysis: Differential forms in ana-lysis, geuIlletry and physics, 2002 51 Y. A. Abramovich and C. D. Allprantls, Problems in operator theory, 2UU2 50 Y. A. Abramovlch and C. D. Aliprantis, An invitation to operator theory, 2002 49 John n.. Harper, Secondary cohomology operatiom;;, 2002 48 Y. Eliashberg and N. Mishachev, Introduction to the h-principle, 2002

47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Clltflslca.\ a.ud quantum computation, 2002 46 JOtieph L. Taylor, !:leveral complex variables with connections to algebraic geometTy A.nd Lie gruups, :lOO~ 45 Inder K. Rana, An introdudiuu to measure and integration, fiecond flditioll.2002

.Tim Agler and John E. MCCarthy, Pick interpola.tion and Hilbert function spaces! 2002 4a N. V. Krylov, Intmrl11C:tion to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crysta.\ bases, 2002 41 Georgi V. Smlrnov, Introduction to t.he t.hl~ry of differential inclusions, 2002 40 Robert E. Greene and St.even G. Krantz, Function theory of aue cumplex variable, '1-1

2U02

39 Larry C. Grove, Classical groups and geomet,Tic algeLcl.L, 2002

38 Elton P. Hsu, Stochast.ir: analysis

011

Ulunifolds, 2002

37 Hershel M. Farkas and Irwin Kra, Theta const.ants, Riemann surfaces and the modu1l'n 36

35 34 33

32 31

group. 2001 Martin Schechter, Principlp.s of funcl.iuuul wlalysis, second edition, 2002 James F. Davis and Paul Kirk, Lecture note8 in algebraic tupology, :lOOI Sigurdur Helgason, Different.ial georucLry, Lie groups, and symmetTic: fipaces, 2001 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A eou.csc in metric geometry, 2001 Robert G. Bartle, A modem t.heory uf integration, 2001 Ralf Korn and Elkc Kurn, Option pricing and pnrt,folio optimization: Modern methods of tinancial mat,hematics, 2001

30 J. C. McConnell and J. C. R.obson, Nuneommutative Noetherian ring6, 2001

29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Note" on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometTy, 2001

26 Rolf Berndt, An introduction

tiD

symplectic geometry, 2001

TITLES 1N THU:i SERIES 21'i 24 23 22 21 20 19 18 17 16 15

Thomas Friedrich, Dirac operators in Riem:'l.nniau gl-'OIIletry, :.!UUO Helmut Koch, Number theory: AIgebmit' numbers and functioWl, 2000 Alberto Candel and T,awrcnce Conlon, Foliations I, 2000 Giinter R. Krause and Thomas H. Lenagan, GrowLh uf algebras and Gclfand-I\irillov dimension, 2000 John B. Conway, A course in operator t.h(,'Ol'y, 2000 Robert E. Gompf and Andras 1. Stipsicz, 4-manifolds and Kirby calculus. 1999 Lawrence C. 'Evans, Purlinl differential equationfl, 1998 Winfrietl J U::lt and Martin Weese, Disc:overiug mudem set theory. 11: Set-theoret.ic: tools fur t'very mathematician, 1997 Henryk Iwaniec, Topics in da.'lsical ttuLulUorphic forms, 1997 Richard V. Kadison and John R. Ringrose, F\mrlamenlals of Lhc Lhcory of operator algebras. Volume TT: Arlvallcl'u thcory, 1Y97 Richard V. Kadison and John R. Ringrose, FllndalllcuLttl~ of the theory of operator algebrAA. Volllll1e T: Elementary theory, 1997

14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergnciic theory uf discrete sample paths, 1996 12 N. V. Krylov, LecLW"l's on elliptic and parabolic equat.ions in Holdcr spaces, 1996 11 Jacques Dixmier, Enveloping algebrAA, 1996 Printing 10 Barry Simon, Representat.ions or finite and compact groups, 199fi 9 Dino Lorenzini, Au iuvitation to arithmetic geometry, 1996 8 Winfricd Just and Martin Weese, DifiCnvering mudt,'m set theory. 1: The basks, 1996 7 Gerald J. Janusz, Alg('lbrak nllmber.' fidds,l;lccond edition, 1996 6 Jens Carsten Jantzcn, Lectures on quantum grouPf\, 1996 .'i Rick Miranda, Algebraic curves and Riema.nn surfuccs, 19!J5 4 RU::lsell A. Gordon, The int.egrals (If LdJ(.'sguc, Denjoy, Perron, and Ilenstnc.k, 1.994 3 WUllam W. Adams Rnd Philippe Loustaunau, An introdlldion t.n Gri..ilmer bases, 1994 2 Jack Graver, Brigitte 8ervatius~ and Herman Servatius, Cumbina.toria.l rigidity~ 1993 1 Ethan Akin, The general topology of dynamical gystern~, 1993