Dimensions of Jet Schemes of Log Singularities

Dimensions of Jet Schemes of Log SingularitiesAuthor(s): Takehiko YasudaSource: American Journal of Mathematics, Vol. 125, No. 5 (Oct., 2003), pp. 1137-1145Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/25099210 .

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DIMENSIONS OF JET SCHEMES OF LOG SINGULARITIES

By Takehiko Yasuda

Abstract. We characterize Kawamata log terminal singularities and log canonical singularities by dimensions of jet schemes. Our main result is Theorem 2.4.

Introduction. For birational geometry, it is natural to consider a pair (X, D) of a normal variety X and a Q-divisor on it, and to consider singularities of

pairs. KLT (Kawamata log terminal) and LC (log canonical) singularities form

important classes of log singularities. They are defined by using a log-resolution and discrepancies. When X is Q-Gorenstein, we can naturally define KLT and

LC also for a pair (X, qY) where F is a closed subscheme of X and q G Q>o In [7], Mustaj? proposed a new point of view in the study of log singularities.

He characterized KLT and LC pairs (X, qY) with X smooth via dimensions of

jet schemes of Y. The aim of this short paper is to extend his result to the case

X is Q-Gorenstein and to shorten his proof. As he did, we also use the motivic

integration as a main tool, which is invented by Kontsevich [5] and extended to

singular varieties by Denef and Loeser [2].

Acknowledgments. This paper forms a part of my master's thesis. First, I

would like to thank Yujiro Kawamata for his encouragement and advice. I also

thank Nobuyuki Kakimi for useful conversation.

1. Motivic integration. In this section, we review the motivic integration invented by Kontsevich [5] and extended by Denef, Loeser [2]. Craw's paper [1] is a nice introduction.

Let k be an algebraically closed field of characteristic zero.

1.1. Extended Grothendieck rings of varieties. We mean by k-variety a

reduced fc-scheme of finite type. The Grothendieck ring of k-varieties, denoted

Ko(Var/k), is the abelian group generated by the isomorphism classes [X] of k

varieties with the relations [X] = [X\Y] + [Y] if y is a closed subvariety of X.

The ring structure is defined by [X][Y] = [X xY]. Let L be the class [A1] of the

Manuscript received January 15, 2002; revised April 18, 2002.

American Journal of Mathematics 125 (2003), 1137-1145.

1137



1138 TAKEHIKO YASUDA

affine line and M the localization ATo(Var/&)[L-1]. Then, define

MQ := M[LM; u G Q] = M[jc,-; i G Zx)]/?*,-)1'

- L).

If we set FWMQ c M9 for each m G Z to be the subgroup generated by elements

[X]LU with dimX + u < -m, then (FmM9)m is a descending filtration with

FmM? FnM9 c Fm+nM?. We define

MQ := limMQ/FmMQ

F?MQ := UmF?MQ/FmMQ.

For a countable set / and ai G M?, i G /, the sum YLiei ai converges if for

any m, {i e I \ at ? FmMQ} is finite.

Definition 1.1. An element a of M? is primitive if either a = 0 or for some

varieties X?, m? G Q and n? G Z,

oo

i>\

such that Xo ^ 0 and dimXo + ?o > dimX? + w,- for every / > 1. The dimension

of a primitive element a is defined to be

dima=i~?? (a = 0)

1 dim Xo + wo (otherwise).

The family of the primitive elements is stable under addition and multiplica

tion, and the following holds: for primitive elements a, ? G M?,

dim a? = dim a + dim ?,

dim (a + ?) =

max{dim a, dim ?},

if dim a > dim/3, then a ? ? is primitive and dim (a

? ?)

= dim a;.

1.2. Jet schemes and motivic measures. Let X be a ^-scheme. For n G

Z>o U {oo}, an n-jet on X is a morphism

Speck[[t]]/(tn+1)-^X,

where we used the convention (t??) = (0) C k[[t]]. The moduli scheme of the

n-jets on X always exists. We call it the n-jet scheme of X and denote it by Ln(X).

If X is of finite type, then for n < oo, so is Ln(X). If X is smooth and of pure



DIMENSIONS OF JET SCHEMES OF LOG SINGULARITIES 1139

dimension d, then, for each n G Z>o, the natural projection Ln+\(X) ?

Ln(X) is

a Zariski locally trivial Ad-bundle.

Now assume that X is a k-variety of pure dimension d. Let 7rn: L^X) ?*

Ln(X) be the canonical projection.

Definition 1.2. A subset A of L^X) is staWe ai level n if we have:

(1) 7rn(A) is a constructible subset of Ln(X),

(2)A =

7T-lnn(A),

(3) for any m > n, the projection 7rm+i(A) ?

7rm(A) is a piecewise trivial

A^-bundle.

(A morphism/: Y ?> X of schemes is called a piecewise trivial ?d-bundle if there

is a finite stratification X = ]JXZ such

that/|^_i(X): f~l(Xi) ?? X,- is isomorphic

to X,- x AJ ?? X/ for each i.) A subset A of Loo(X) is staWe if it is stable at level

n for some n G Z>o. We see that the stable subsets of Loo(X) constitute a Boolean algebra, that is,

stable under finite intersection and finite union. For a stable subset A of L^X), the element [7rn(A)]L~nd G M is independent of the choice of n >> 0. (We can

define the class of a constructible set in M and M in the evident fashion.) So the

map

px' {stable subsets of L^X)} ? M

A h-> [7Tn(A)]h-nd (n > 0)

is a finite additive measure. We can extend px to the family of the measurable

subsets of LqqX, which is a family "big enough." For details, see [3], [6]. We

call px the motivic measure on L^X. Let A c LooCÔ be a measurable subset and v\ A?>Qu{oo}a function. We

say that v is a measurable function if the fibers are measurable and px(v~l(oc)) =

0.

Definition 1.3. For a measurable function v, we formally define the motivic

integral of L^ by

[h^dpx:=J2^^-l(u))hu. JA u?Q

If the infinite sum converges in M9, then this is well defined as an element of

M9.

Let Y c X be a closed subscheme and o its ideal sheaf. A closed point

7 G Loo(X) corresponds to a morphism 7': Spec?[|>]] ? X. The function

FY: Loo(X) -

Z>o U {oo}

7 ^ nif (7/)"1a =

(iw)




is a measurable function. For a Q-divisor D = J2i Qiî on X with A a prime

divisor, we define a measurable function Fd := YîQi^Dr The following explicit

formula gives a way to compute motivic integrals.

Lemma 1.4. Assume that X is smooth and Y,Ui A is a SNC (simple normal

crossing) divisor on X with D? aprime divisor. Let m? G Z>o, (1 < / < s), and put J := {i | mi > 0} C {1,..., s}. Then we have

?x m^V/))

= [D?j]QL

- l)lylL-S^,

where

D?j-=nD\ u a. ieJ x

ie{i,...,s}\j

Proof. See [1, the proof of Thm. 2.15].

2. Main theorem. Let X be an irreducible and normal variety of dimension

d. Assume that X is Q-Gorenstein, that is, for some r G Z>o, rKx is a Cartier

divisor. For a resolution p: X ?> X, the relative canonical divisor Kîx is

Kx/x-=-r(rKx-p*(rKx)).

Let 7 be a closed subscheme of X and a its ideal sheaf. By Hironaka's

theorem, there are a resolution p: X ?> X and a SNC divisor ]T?=1 ?>/ on X such

that:

p_1a =

Ox( -

E/J/A) for some yt G Z>0,

^ := ^x/x

= S/ a/A for some a? G Q.

Fix this notation through the rest of the paper. In the proof of Theorem 2.4, we

put additional conditions on p.

Definition 2.1. For q G Q>o> we say that the pair (X,qY) is KLT (Kawamata

log terminal), resp. LC (log canonical) if for every /, ?qyi + a? + 1 > 0, resp.

-qyi + a,- + 1 > 0.

Let Xreg be the smooth locus of X and ?: Xreg e~> X the inclusion. Let Q^

be the d-th exterior power of the sheaf of differentials over X. Then u^ =

?*((Q^)(g)r|xreg) is an invertible sheaf. We define an ideal sheaf J C Ox by the

following equation:

,74rl= Image ((??)?r->4rl).




Let Z C X be the closed subscheme associated to J. Then SuppZ =

XSing. The

following is a variation of the transformation rule, a key of the proof of the main

theorem.

Theorem 2.2. Let A c L^X be a measurable subset and v. A ? Q U {00} a

measurable function. Then we have the following equality:

[ L^+(1/r)'Fz dpx = [ L^??-^ dux.

Ja JpîA)

Proof It is a direct consequence of the transformation rule [2, Lern. 3.3]. D

For each e G Z>o, we put Ae := F^l(e). For each n,e G Z>o, we define

Len(Y):=Ln(Y)n>irn(Ae),

where we take the intersection in Ln(X). Of course, Len(Y) depends on the inclusion

F CX.

Lemma 2.3. There is a positive integer 6 such that for any n, e G Z>o with

n > 0e, the natural projection 7rn+\(Ae) ?> 7rn(Ae) is a piecewise trivial ?d-bundle.

Proof See [2, Lern. 4.1].

Theorem 2.4. Let 9 be a positive integer as in Lemma 2.3. Suppose Supp Y D

Xsing and take l G Z>o with a1 C Je. Let IY be the closed subscheme ofX associated

to the ideal sheaf a1. Then,

(1) (X, qY) is KLT iff for any e,n<E Z>0 with n+l>6e,

dimL^(/y) + e/r <(n+ \)(d -

q/l).

(2) (X, qY) is LC iff for any e, n G Z>o with n+l > 9e,

dimL^(ZF) + e/r <(n+ \)(d -

q/l).

Proof We prove only (1). We may assume / = 1 by replacing Y with IY and q with q/l. When Ae = 0, the inequality in the theorem trivially holds. So suppose

Ae j? 0. This means Len(Y) j? 0 for every n. For each n G Z>0, let Bn := Fyl(n)

and B>n := Fy xdl>>n). For each e, n G Z>o with n+l > Oe, consider the following element of MQ:

S(e,n) := / L^+OA)^^ JBn+lnAe

= fix(Bn+lnAe)^n+l^+e/\




By Lemma 2.3,

S(e,n) = (px(B>n+i HAe)

- px(B>n+2 nAe))L(n+l)?+e/r

= ([Len(Y)]

- [Len+l(Y)]L-d)L-nd+(n+l)q+e/r.

By Lemma 2.3 again, we have dimL*+1(F) < dimLen(Y) + d. Unless the equality holds, then S(e, n) is a primitive element and

(2.1) dimS(??,n) =

dimL^(F) - nd + (n + \)q + e/r.

In fact S(e, n) is always a primitive element as we see below.

Now we put addtional conditions on the resolution p. Suppose that y? > 0 and p~lJ

= Ox(

- YÛi *iA) with n > 0. Such p exists from the assumption

Supp Y D Xsing. We set Y := J2?=\ y/A and Z := ?-=1 ZiDt. From Theorem 2.2,

we have

S(e,n)= ?\?FY-F*dpx

where the domain of the integration is

Ffl(n+l)nF~l(e).

From Lemma 1.4, we obtain

S(e,n) = Yl Yl lD?jKh

- l)|/|L"S(-?y/+fl/+1>?/,

JC{l,...,s}m M

where

M = M(J,n,e)

:= jm

= (mi)ieJ G (Z>0)7 | J^ym = n + 1 and ̂ z?m?

= e}

.

When D?j ji 0, [D?j]?L- l^L" S (-?y/+?.-+i>?i is a primitive element of dimension

d ? J2ieJ (

~~ SW + a* + ^)mi- Hence, S(e, n) is also primitive and

(2.2) dim S(e, n)- sup ( d - Y] ( -

qyt + a? + l)m,- I . JC{l,...,s}, D?& \

. /

mGAf

"0w/;y i/" part. The proof is by contradiction. So assume that (X, qY) is KLT, that is, for every i, ?qyi + a,- + 1 > 0, and that for some n', e' G Z>o with




ri + 1 > de',

dimZ^(F) + e'/r > (nf + \)(d -

q).

By (2.2), we have dimS(e', n') < d (even if S(e, n) = 0 by the convention dimO =

?oo). On the other hand, since

dimL*',(F) - rid + (ri + \)q + e'/r > d,

the equality in (2.1) does not hold. Hence dim L*,+1(F) = dim L*',(F) + d and

hence dimL*',+1(F) + e'/r > ((n' + 1) + \)(d

- q). By the same argument, we have

dimL*'+2(F) =

dimL*'+1(F) + d and so on. This contradicts Lemma 2.5.

"If "

part. From inequality (2.1), we deduce that for any e, n with n +1 > de,

(2.3) dim S(e,n) <d.

This is true even if dimL^+1(F) = dimL*(F)+d. By assumption, we have On < yi

for every i. When n = y i ? 1 and e = zu the set M({i}, yi

? \,zd contains the

element m = (m/

= 1) for every /. Therefore, in view of (2.2) and (2.3), we obtain

that for any /, ?qyi + ai + 1 > 0, that is, (X, qY) is KLT. We have thus completed the proof. D

Lemma 2.5. For every e G Z>o, there is a real numbers b < d such that for

every n G Z>o,

dim Len(Y)<bn + (const).

Proof. It suffices to find a strictly increasing linear function ip: Z>o ?> Z>o and a real number b < d such that

dimLe^(n)(Y) < btp(n) + (const).

First, the case Y reduced: For m,n G Z>o with m > n, we denote by 7r^ the

natural projection Lm(X) ?? L?(X). By Greenberg's theorem [4, Cor. 1], there is

a linear function g: Z>o ?? Z>o such that:

for every n G Z>o, g(ft) > n,

TTadoodO) =

7rf(w)(Lg(n)(7)).

Since L?(n)(Y) C (Tr^Wf^^/F)),

dimL^?F) < dim(^r^^L^F)).




By the definition of g,

7r^\Leg(n)(Y)) =

7rn(Le00(Y)).

Let d' be the dimension of F. By [2, Lern 4.3],

dimTrf^L^F)) =

dim7r?(Z4(F)) < (n + \)d'.

Hence, by Lemma 2.3,

dimL^F) < (n+l)d' + (g(n)-n)d =

g(n)d + n(d' ?

d) + (const)

= g(n)(d + (d' -

d)/c) + (const),

where c is the constant such that g(n) = cn + (const). We have thus proved the

assertion in this case.

The general case: It suffices to show the case where F = /(Fred) for some

/ G Z>o- By definition, we have the following

Leln_x(Y) =

7rln^(Fyl(ln)nAe),

Len^(YTed) =

7rn^(FyTld(n)nAe).

Because for n, e with n > 6e, Fyl(ln)C\Ae =

FyXJ(n) f)Ae is stable at level n, we

have Leln_x(Y) =

(TTl//Z^)-lLen(YVQ?). Therefore, for some b < d,

codim(Lf?_1(F)/7r/?_i(A??)) =

codim(L^_1(Fred)/7rw_i(Ae))

> (d ?

b)n + (const).

Because dim7rn(A^) = dn + (const), we have

dimLeln_x(Y) < din - (d

- b)n + (const)

< (d-(d- b)/l)(ln -

1) + (const).

This completes the proof.

Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro,

Tokyo, 153-8914, Japan

E-mail: [email protected]




REFERENCES

[1] A. Craw, An introduction to motivic integration, preprint, arXive:math.AG/9911179. [2] J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent.

Math. 135 (1999), 201-232.

[3] _, Motivic integration, quotient singularities and the McKay correspondence, Compositio Math.

131 (2002), 267-290.

[4] M. J. Greenberg, Rational points in Henselian discrete valuation rings, Inst. Hautes ?tudes Sei. Publ.

Math. 31 (1966), 59-64.

[5] M. Kontsevich, Lecture at orsay, 1995.

[6] E. Looijenga, Motivic measures, S?minaire Bourbaki, vol. 1999/2000, Ast?risque 276 (2002), 267-297.

[7] M. Mustaj?, Singularities of Pairs via Jet Schemes, J. Amer. Math. Soc. 15 (2002), 599-615.



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Dimensions of Jet Schemes of Log Singularities