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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 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
[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
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)
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
[ 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)
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
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.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;' 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 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 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.., 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
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,
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' 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
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
'
= (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
' : 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;)
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....)
..,.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,
(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 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 =
0, Zj-1 ,
i.= O.
Thus, by the Leray spectral sequence,
(6.51)
Rigtlr
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 '"'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')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,
° rand
Thus 2 follows if r
oX
< r < k, then,~ ~ < 0,
A:.
Suppose that k = rand (
i
+- (~ __ ~)_a , j
°
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 - 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 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 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 ] 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
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
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}
d
where Ui(X, y) E K[x, y] for all i, the ai(x, y)z 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)
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
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 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 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 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
Bibliography
[1] Abhyankar 1 S., Lucal1.l/niformization on algebraic ,mrfaces over gr'Q'Uud fields of ChQ,1'acte7"istic pI:- 0, Anna,lf; of Math, 63 (1956), 491 526.
[21 Abhyankar, S. Simultaneous n:sul'ution f01'
alg~braic !mrfar..es,
Amer. J. Math 78
(1956),761-790.
{3] Abhyankar, S., Ilamification theoretic
method~' iT~ algeb7uic geometry, AnnalR of
Math.
Studies, 43 1 Princeton University Press, Princeton, N.I, 1959.
[4] A1Jhyanka:r, S., Resolution of Flingttlarities of embedded algcbmic surfaces, second edition, Springer Verlag, New York, Berliu, Heidelberg, 1998. [5] Abhyankar, S., Desingularization of pla,ne curves, AMS Proceedings of Sylllp. ill PW'e Math. 40, part. 1 (198:i), 1-45. [6) Abhyankar, S., A 1gebmie geometry for scie.nti,st.'i and engineers, AMS Mathellll:l.Lical Swvcys and Monographs 35, Amer. Math. Soc., Providence, RI, 1990.
[7) Abhyankar, S., Good points of a hypenu7jace, Advances in Math. 68 (19XX), 87-256, [8) Abhyankar, S., Resolut'ion of singularities and modular Galozs theory, Bulletin of the AMS 38 (2000), 131-171. [9] Abramovich, D. and de Jong, A.J., Smoothness, se.mistability and toroidal .qeometry, Journal of Algebraic Geometry, 6 (1997), 789-801.
110] Abramovich, D. and Karu, K., Weak semistablc
7Y~d'Uction in
characte.r1..'1tic 0, Invent.
Math., 139 (2000), 241·"273. [11] Abramovich, D., Karu, K., MatFluki, K., Wlodarczyk, J., Tu1'ification and factorization of bimtiortal maps, J. Amer. Math. Soc. 15 (2002),351-572.
[12) Albanese,G, Tra1£sfonnazione birazionale. di una Ruperfic'i.e al.qebrica qualunquc 'in un'altra priva di punti multipli, Rend. Circ. Mat. Palerrno 48 (1924), 321-332. {13] Atiyah, M.F. and Macdonald, I,G., Introduction to r.omm~Ltatitle algebra, AddisonWesley, Reading, MA (1969). [14J Bierstone, E. and Milman, P, Canonzcal deai1lgular";'zation in characteristic zero by blo1lJingup the maximal strata of a local invariant, Inv. Mat.h 128 (1997), 207 302. [15] Bodnar, G. and Sehkho, J., A computer algorithm for the 7'esolution of singularities, in Resolution of Singularities, (Obergurgl, 1977), Progr. Math 181, BirkhauFler, Ba.'lel, 2000, 53-80.
179
Bibliogl'apby
180
[16] Bogomolov F. and Pantev, T., Weak Hi7'Onaka thco'rem, ~faUlCllLULical Re~euf(;h Let-
ters 3 (1996), 299-Ml7. II7] Bravo, A., Encinas, S., VillulIlliyor, 0., A simpi'tjied proof of desm.qularization and applimtirl'n,'i, to 8.PP8a.r in Revista r..1atematica Ibcroamcricauo.. 118J Bril'Bkoru, E. unu Kuorrer, H., Plane algebral,c CUTtl e8, Birkhallser, BMel 1 11)/:0\6. [19) Cano, II'., Desinglnl'h:ation of plane t:ector fields, Trans. A1UUr. M:alh. SO(:. 296 (1986L 83 93. [20] Cano, F., Desiny'ulu''''lzat'iun slmtcgic8 for three-dimensional vector .fields, Lecture Notes in Mathema.tks 1259, Springer-Verlag, Heidelberg, 1987. 1211 Cauu, F., Reduction of the szn.qularztzes of foliation.'i and applir.{J.t.ion,.., Singulari t.ie8 Symposium, Banach Center. Publ. 44, Polish Acuu. Sci., ':Vl:I.csaw, 1998,51 71. [22] Chevalley. C., Intmdur,tion to the thp.orll of a.l.gp.hmic j1met.ions of one variable, AM8, J..!Iathematicn.I Survc,ys 6, AlUCl'. Malh. Soc., New York! 1951. [2:i] Cossart., V,. j)f',8ingnta.dza,f.1:on of embedded excellent surjaces, Tohoku r\'la.lh. .]ouru. 33 (1981), 25 33. [2/1] C08san. V.. Polyedre camcU1"ist'ique d',uw.: ~finY'ulU'I'il{f, Thesis, Uuiver:;iLe de ParisSud! Centre rl'Orsay, HJK7.
(25J Cutkosky, S.D.. Local JUdo7'izatwn of bnYltwnal m.ap,';, AdvanceR in Math. 132, (1 !-}!-J 7),
12GI
l() 7-:3 l.j.
Cutko~ky, S.D.,
Local rnonomzalzzation and fa(;torizatinTl. of mnrphi... m.,o;• .~st,erisqlle
260, (1999). [27] Cmkosky, S.D., Mmwmin.liza.tin17. nf morphisms from 9jolds to s'U1jlLces, L(duw Kotes in Mut.h 1786, Springer Verlag, Berlin, Heidelberp;, New York, 2U1I2. [28] Cutkosky. S.D., '"ora,1. rnonmnio.lizo.tio/l. of transcendental extens'ions, pl'cpdul.
129] Cutkosky, S.D., Simultaneous 'resolutton of sm,qulaMtzes. Proc. Anl(~I'ir.an r..1at.h. Soc.. 128 (2000). 1905-1910. 130] Cutko:;ky, S.D., Gener'icallll finitfl mO'lphi,.. mll a,nd liim,1tltnneOll..r:, re.sol1f,tion of sing'U larities. COllt.elllporary YIaLhcl1JUt.ics 331 (2003), 75 99. [:31] Cutkosky, S.D, anrl He.l·7,Og, ,I., Cohen-ll1acattLay coordinate 7'ings of blowup schemes. COlJUllcnl. !vlaLIt. Relv. 72 (1997), 605 617. [~i2] Clltkosky, S.D. and Ghezzi, L" Completions oJ'!.:al,tta.l'ic)'fL 'I'i'llys, La appcur iu Conlt'1JJ-
IJurary Ivltt.thematics. [:J3] Cut kosky, S.D. and Knshdl('~'evu. 0., Jt.lunu·r"'lal'izat'ion oJ strongly prepared morphtsms from rnon..'1ingnlar N-fold.... t.o $n'rfaces. t.o appear in J. AIg, [34] Cutkosky, S.D. ami PilLuul, 0., Monormal resolutwns of morphi.9m,f1 of algebra.ir:. .'mrface,.., COl11nlllnk.."l.tions in Algebra 28 (2000)! 503G--G95D.
/351 CllLko~ky, S.D. and Piltant, 0 .. Ramilicn.tion of ?!n.lu.ation.r;, to Hppp.ar in Arlva.nces ill ;\:Jat.hematics 183, (2004), 1-79.
[36] de .Tong, A. .T., 5'm,oothnp.....'i, (109G), 51 93.
,~P.m.i.."tfl,hility
and (1,Ltemtions, Publ. Math. I.H.E.S. 83
[:17] 1';iJ'p.nhnd, D., Commuta'uue algcbm 'With u
I1i(~'W tu'Wunls ulyeb'fv:ie yeomctry,
Grau,
Texts in Math. 15U, Springer Verlag, IIp.idelberg! Uerlin, New York, 1995.
[:38) Eisenbud, D. and Ha.rris, J., Tlte ycumety·y oJ schemes, Grad. Texts in Math. 197, Springer Verlag, liflidelbp.l·g, Berlin, I\ew York. 2000.
[39]
Euciu~,
S., Haus~l', H., Stron.Q resolutzon of sin,q1J,laritie.'1 in (;haradp.ri.'itir. zp.m, Comment I'v1at h. Ileiv. 77 (2002), 82I-84u.
un
Bi bliograpb,y
[40] Encinas S. amI Vilhunayor 0., A umr,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
