Dimensions of jet schemes of log singularities

7 Pages·0.14 MB·

by  Takehiko Yasuda

DIMENSIONS OF JET SCHEMES OF LOG
SINGULARITIES
2
0
0
TAKEHIKO YASUDA
2

n
a
J

5 Abstract. We characterize Kawamata log terminal singularities and
1 logcanonicalsingularitiesbydimensionsofjetschemes. Ourmainresult
is Theorem 2.4.
]
G
A Introduction
.
h For birationalgeometry, itisnaturaltoconsiderapair(X,D)ofanormal
t
a variety X and a Q-divisor on it, and to consider singularities of pairs. KLT
m
(Kawamata log terminal) and LC (log canonical) singularities form impor-
[ tant classes of log singularities. They are defined by using a log-resolution

1 and discrepancies. When X is Q-Gorenstein, we can naturally define KLT
v and LC also for a pair (X,qY) where Y is a closed subscheme of X and
9 q ∈ Q .
2 >0
In [Mus01], Musta¸t˘a proposed a new point of view in the study of log
1
1 singularities. He characterized KLT and LC pairs (X,qY) with X smooth
0 via dimensions of jet schemes of Y. The aim of this short paper is to extend
2
his result to the case X is Q-Gorenstein and to shorten his proof. As he
0
/ did, we also use the motivic integration as a main tool, which is invented by
h
Kontsevich [Kon95] and extended to singular varieties by Denef and Loeser
t
a [DL99a].
m
:
v Acknowledgements. Thispaperformsapartofmymaster’sthesis. First,
i I would like to thank Yujiro Kawamata for his encouragement and advise.
X
I also thank Nobuyuki Kakimi for useful conversation.
r
a
1. Motivic integration
In this section, we review the motivic integration invented by Kontsevich
[Kon95] and extended by Denef, Loeser [DL99a]. Craw’s paper [Cra99] is a
nice introduction.
Let k be an algebraically closed field of characteristic zero.
1.1. Extended Grothendieck rings of varieties. TheGrothendieck ring
of k-varieties, denoted K (Var/k), is the abelian group generated by the
0
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] =
1
2 TAKEHIKO YASUDA
[X×Y]. LetLbetheclass[A1]oftheaffinelineandletMbethelocalization
K (Var/k)[L−1]. We define the dimension of an element of M by
0
dim( [X ]Lmi):= max(dimX +m ),
i i i
i
Xi
with the convention dim∅:= −∞. Then, the following hold: for α, β ∈ M,
• dimαβ =dimα+dimβ,
• dim(α+β) ≤ max{dimα,dimβ} and the equality holds if dimα 6=
dimβ.
If we set F M := {α ∈ M|dimα ≤ m}, then (F M) is an ascending
m m m
filtration with F M·F M ⊂ F M. We define the complete Grothendieck
m n m+n
ring of k-varieties by
Mˆ := limM/F M (m → −∞).
m
←−
Any element α of Mˆ is expressed as α = α such that
m∈Z m
• α = 0 or dimα = m, P
m m
• α = 0 for m ≫ 0.
m
Then we define the dimension of α to be max{m|α 6= 0}.
m
We denote by MˆQ the ring Mˆ[Lq;q ∈ Q]. Any element of MˆQ is expressed
as αLq with α ∈ Mˆ, q ∈Q. We definethe dimension of αLq to bedimα+q.
1.2. Jet schemes and motivic measures. Let X be a k-scheme. For
n∈ Z ∪{∞}, an n-jet on X is a morphism
≥0
Speck[[t]]/(tn+1) → X,
whereweused the convention (t∞) = (0) ⊂ k[[t]]. Themodulischeme of the
n-jets on X always exists. We call it the n-jet scheme of X and denote it by
L (X). If X is of finite type, then for n < ∞, so is L (X). If X is smooth
n n
and of pure dimension d, then, for each n ∈ Z , the natural projection
≥0
L (X) → L (X) is a Zariski locally trivial Ad-bundle.
n+1 n
NowassumethatX isak-variety ofpuredimensiond. Letπ :L (X) →
n ∞
L (X) be the canonical projection.
n
Definition 1.1. A subset A of L (X) is stable at level n if we have:
∞
(1) π (A) is a constructible subset of L (X),
n n
(2) A= π−1π (A),
n n
(3) foranym ≥ n,theprojectionπ (A) → π (A)isapiecewisetrivial
m+1 m
Ad-bundle.
(A morphism f : Y → X of schemes is called a piecewise trivial Ad-bundle
if there is a stratification X = Xi such that f|f−1(Xi) : f−1(Xi) → Xi is
isomorphic to X ×Ad → X fo`r each i.) A subset A of L (X) is stable if
i i ∞
it is stable at level n for some n ∈ Z .
≥0
We see that the stable subsets of L (X) constitute a Boolean algebra,
∞
that is, stable under finite intersection and finite union. For a stable subset
DIMENSIONS OF JET SCHEMES OF LOG SINGULARITIES 3
A of L (X), an element [π (A)]L−nd ∈ Mˆ is independent of the choice of
∞ n
n ≫ 0. (We can define the class of a constructible set in M and Mˆ in the
evident fashion.) So the map
µ : {stable subsets of L (X)} → Mˆ
X ∞
A 7→ [π (A)]L−nd (n ≫ 0)
n
is a finite additive measure. We can extend µ to the family of the mea-
X
surable subsets of L X, which is a family ‘big enough’. For details, see
∞
[DL99b], [Loo00]. We call µ the motivic measure on L X.
X ∞
Let A ⊂ L (X) bea measurable subsetand ν :A→ Q∪{∞}a function.
∞
We say that ν is a measurable function if the fibers are measurable and
µ (ν−1(∞)) = 0.
X
Definition 1.2. Forameasurablefunctionν,weformallydefinethemotivic
integration of Lν by
Lνdµ := µ (ν−1(n))Ln.
X X
Z
A nX∈Q
We say that Lν is integrable if this infinite sum converges in MˆQ.
Let Y ⊂ X be a closed subscheme and a its ideal sheaf. A closed point
γ ∈ L (X) corresponds to a morphism γ′ : Speck[[t]] → X. The function
∞
F :L (X) → Z ∪{∞}
Y ∞ ≥0
γ 7→ n if (γ′)−1a = (tn)
is a measurable function. For a Q-divisor D = q D on X with D
i i i i
a prime divisor, we define a measurable functionPFD := iqiFDi. The
following explicit formula give a way to compute motivic intPegrations.
Lemma 1.3. Assume that X is smooth and s D is a SNC divisor on
i=1 i
X with Di a prime divisor. P
Let m ∈ Z , (1 ≤ i ≤ s), and put J := {i|m > 0} ⊂ {1,...,s}. Then
i ≥0 i
we have
µ ( F−1(m )) = [D◦](L−1)|J|L− mi,
X Di i J P
\i
where
D◦ := D D .
J i(cid:31) i
i\∈J i∈{1,[...,s}\J
Proof. See [Cra99, the proof of Thm. 2.15].
4 TAKEHIKO YASUDA
2. Main theorem
LetX beanormalvarietyofdimensiond. AssumethatX isQ-Gorenstein,
that is, for some r ∈ Z , rK is a Cartier divisor. For a resolution
>0 X
p :X˜ → X, the relative canonical divisor K is
X˜/X
1
K := (rK −p∗(rK )).
X˜/X r X˜ X
Let Y 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 D on X˜
i i
such that P
• p−1a = O (− y D ) for some y ∈ Z ,
X˜ i i i i ≥0
• K := KX˜/X =P iaiDi for some ai ∈ Q.
Fix these notations thPrough the rest of the paper. In the proof of Theorem
2.4, we suppose additional conditions on p.
Definition 2.1. For q ∈Q , we say that the pair (X,qY) is KLT (Kawa-
>0
mata log terminal), resp. LC (log canonical) if for every i, −qy +a +1 > 0,
i i
resp. −qy +a +1 ≥ 0.
i i
Let X be the smooth locus of X and ι :X ֒→ X the inclusion. Then
reg reg
ω[r] = ι ((Ωd )⊗r) is an invertible sheaf. We define an ideal sheaf J ⊂ O
X ∗ X X
by the following equation:
Jω[r] = Image((Ωd )⊗r → ω[r]).
X X X
Let Z ⊂ X be the closed subscheme associated to J. Then SuppZ = X .
sing
The following is a variation of the transformation rule, a key of the proof of
the main theorem.
Theorem 2.2. Let A ⊂L X be a measurable subset and ν : A→ Q∪{∞}
∞
a measurable function. Then we have the following equality:
Lν+(1/r)·FZdµ = Lν◦p∞−FKdµ .
ZA X Zp−∞1(A) X˜
Proof. It is a direct consequence of the transformation rule [DL99a, Lem.
3.3].
For each e∈ Z , we put A := F−1(e). For each n,e∈ Z , we define
≥0 e Z ≥0
Le(Y) := L (Y)∩π (A ),
n n n e
where we take the intersection in L (X). Of course, Le(Y) depends on the
n n
inclusion Y ⊂ X.
Lemma 2.3. There is a positive integer θ such that for any n,e∈ Z with
≥0
n ≥ θe, the natural projection π (A ) → π (A ) is a piecewise trivial
n+1 e n e
Ad-bundle.
Proof. See [DL99a, Lem. 4.1].
DIMENSIONS OF JET SCHEMES OF LOG SINGULARITIES 5
Theorem 2.4. Suppose SuppZ ⊃ X . Let θ be a positive integer as in
sing
Lemma 2.3. For l ∈ Z , let lY be the closed subscheme of X associated to
>0
the ideal sheaf al. Suppose that al ⊂ Jθ. Then,
(1) (X,qY) is KLT iff for any e,n ∈ Z with n ≥ θe,
≥0
dimLe(lY)+e/r < (n+1)(d−q/l).
n
(2) (X,qY) is LC iff for any e,n ∈ Z with n ≥ θe,
≥0
dimLe(lY)+e/r ≤ (n+1)(d−q/l).
n
Proof. We prove only (1). For each n ∈Z , let B := F−1(n) and B :=
≥0 n Y ≥n
F−1(Z ). For each e,n ∈ Z with n≥ θe, consider the following element
Y ≥n ≥0
of MˆQ:
S(e,n) := LqFY+(1/r)·FZdµ
X
Z
Bn+1∩Ae
= µ (B ∩A )L(n+1)q+e/r.
X n+1 e
By Lemma 2.3,
S(e,n) =(µ (B ∩A )−µ (B )∩A ))L(n+1)q+e/r
X ≥n+1 e X ≥n+2 e
=([Le(Y)]−[Le (Y)]L−d)L−nd+(n+1)q+e/r.
n n+1
By Lemma 2.3 again, we have dimLe (Y) ≤ dimLe(Y)+d, and hence
n+1 n
(2.1) dimS(e,n) ≤ dimLe(Y)−nd+(n+1)q+e/r.
n
If the equality does not hold, then dimLe (Y) = dimLe(Y)+d.
n+1 n
We suppose that p−1J= O (− z D ) with z >0.
X˜ i i i i
We set Y˜ := y D and Z˜ := P z D . From Theorem 2.2, we have
i i i i i i
P P
S(e,n) = LqFY˜−FKdµ
Z X˜
where the domain of the integration is
F−1(n+1)∩F−1(e).
Y˜ Z˜
From Lemma 1.3, we obtain
S(e,n) = [D◦](L−1)|J|L− (−qyi+ai+1)mi,
J
P
J⊂{X1,...,s}mX∈M
where
M = M(J,n,e)
:= {m= (m ) ∈ (Z )J | y m = n and z m = e}.
i i∈J >0 i i i i
X X
For each J and m, the highest term of
[D◦](L−1)|J|L− (−qyi+ai+1)mi
J
P
6 TAKEHIKO YASUDA
has a positive coefficient. Hence,
(2.2) dimS(e,n) = max (d− (−qy +a +1)m ).
i i i
J⊂m{1∈,M...,s} Xi
‘Only if’ part: The proof is by contradiction. So assume that (X,qY) is
KLT, that is, for every i, −qy +a +1 > 0, and that for some n′,e′ ∈ Z
i i ≥0
with n′ > θe′,
dimLen′′(Y)+e′/r ≥ (n′+1)(d−q).
By (2.2), we have dimS(e′,n′) < d. On the other hand, since
dimLen′′(Y)−n′d+(n′+1)q+e′/r ≥ d,
the equality in (2.1) does not hold. Hence dimLe′ (Y) = dimLe′(Y)+d
n′+1 n′
andhencedimLe′ (Y)+e′/r ≤ ((n′+1)+1)(d−q). Bythesameargument,
n′+1
we have dimLe′ (Y)= dimLe′ (Y)+d and so on. It contradicts Lemma
n′+2 n′+1
2.5.
‘If’ part: From inequality (2.1), we have that for any e, n with n ≥ θe,
(2.3) dimS(e,n) < d.
In view of (2.2) and (2.3), it is easy to find that for any i, −qy +a +1 > 0,
i i
that is, (X,qY) is KLT. We have thus completed the proof.
Lemma 2.5. Then for each e ∈ Z , there is a real numbers b such that
≥0
b< d and for every n ∈ Z ,
≥0
dimLe(Y)≤ bn+(const).
n
Proof. It suffices to find an increasing linear function ψ : Z → Z and a
≥0 ≥0
real number b< d such that
dimLe (Y)≤ bψ(n)+(const).
ψ(n)
First, the case Y reduced: For m,n ∈ Z with m ≥ n, we denote by πm
≥0 n
the natural projection L (X) → L (X). By Greenberg’s theorem [Gre66,
m n
Cor. 1], there is a linear function g :Z → Z such that
≥0 ≥0
• for every n ∈ Z , g(n) ≥n,
≥0
g(n)
• π (L (Y)) = π (L (Y)).
n ∞ n g(n)
Since Le (Y) ⊂ (πg(n))−1πg(n)(Le (Y)),
g(n) n n g(n)
dimLe (Y) ≤ dim(πg(n))−1πg(n)(Le (Y)).
g(n) n n g(n)
By the definition of g,
πg(n)(Le (Y)) = π (Le (Y)).
n g(n) n ∞
Let d′ be the dimension of Y. By [DL99a, Lem 4.3],
dimπg(n)(Le (Y)) = dimπ (Le (Y)) ≤ dim(n+1)d′.
n g(n) n ∞
DIMENSIONS OF JET SCHEMES OF LOG SINGULARITIES 7
Hence, by Lemma 2.3,
dimLe (Y)≤ (n+1)d′ +(g(n)−n)d+(const)
g(n)
= g(n)d+n(d′−d)+(const)
= g(n)(d+(d′−d)/C )+(const),
1
where C is the constant such that g(n) = C n + (const). We have thus
1 1
proved the assertion in this case.
The general case: It suffices to show the case where Y = l(Y ) for some
red
l ∈ Z . By the definitions, we have the following
>0
Le (Y)= π (F−1(ln)∩A ),
ln−1 ln−1 Y e
Le (Y )= π (F−1 (n)∩A ).
n−1 red n−1 Yred e
Because for n,e with n ≥ θe, F−1(ln)∩A = F−1 (n)∩A is stable at level
Y e Yred e
n, we have Le (Y) = (πln−1)−1Le(Y ). Therefore, for some b< d,
ln−1 n−1 n red
codim(Le (Y)/π (A ))= codim(Le (Y )/π (A ))
ln−1 ln−1 e n−1 red n−1 e
≥ (d−b)ϕ(n)+(const).
Because dimπ (A ) = dn+(const), we have
n e
dimLe (Y) ≤ dln−(d−b)n+(const)
ln−1
≤ (d−(d−b)/l)(ln−1)+(const).
This completes the proof.
References
[Cra99] Alastair Craw. An introduction to motivic integration. preprint,
math.AG/9911179, 1999.
[DL99a] JanDenefandFran¸coisLoeser.Germsofarcsonsingularalgebraicvarietiesand
motivic integration. Invent. Math., 135(1):201–232, 1999.
[DL99b] Jan Denef and Fran¸cois Loeser. Motivic integration, quotient singularities and
theMcKay correspondence. preprint, math.AG/9903187, 1999.
[Gre66] MarvinJ.Greenberg.RationalpointsinHenseliandiscretevaluationrings.Publ.
Math. I.H.E.S.,31:59–64, 1966.
[Kon95] Maxim Kontsevich.Lecture at orsay. 1995.
[Loo00] Eduard Looijenga. Motivic measures. preprint, math.AG/0006220, 2000.
[Mus01] Mircea Musta¸t˘a. Singularities of Pairs via Jet Schemes. preprint,
math.AG/0102201, 2001.
DepartmentofMathematicalSciences,UniversityofTokyo,Komaba,Meguro,
Tokyo, 153-8914, Japan
E-mail address: [email protected]