Singularities of log varieties via jet schemes

0.2 MB·English

by  Kalle Karu

SINGULARITIES OF LOG VARIETIES VIA JET SCHEMES
2
1
0 KALLEKARUANDANDREWSTAAL
2

n Abstract. We study logarithmic jet schemes of a log scheme and generalize a
a
theorem of M. Musta¸t˘a from the case of ordinary jet schemes to the logarithmic
J
case. IfX isanormallocalcompleteintersectionlogvariety,thenXhascanonical
1 singularities if and only if thelog jet schemes of X are irreducible.
3

]
G
A 1. Introduction
.
h
M. Musta¸t˘a proved in [9] that a normal, local complete intersection variety X has
t
a canonical singularities if and only if the jet schemes J (X) are irreducible for all m.
m
m
The goal of this article is to generalize this result to the case of log varieties.
[
We follow K. Kato’s terminology (cf. [5]) and consider fine and saturated log

1 schemes X = (X,M ), where X is a scheme and M a sheaf of monoids giving
X X
v
the log structure on X. We call X a log variety if X is a variety. The simplest log
6
4 structures on X are associated to open embeddings U ⊂ X, with
6
M = {f ∈ O |f is invertible on U}.
6 X X
.
1 Special cases of this include the trivial log structure U = X, toric varieties T ⊂ Y,
0 and toroidal embeddings U ⊂ X.
X
2
1 The log jet scheme Jm(X) of the log scheme X was constructed in full generality
: by S. Dutter in [1]. (The special case where the log structure on X is associated to
v
i an open embedding U ⊂ X was also worked out in [11].) The log jet scheme Jm(X)
X
is again a log scheme. Its S-valued points are the S-valued log jets in X:
r
a Hom (S,J (X)) ≃ Hom (S×j ,X).
log.sch m log.sch m
Here S is any log scheme, j = Speck[t]/(tm+1) with trivial log structure M =
m jm
O∗ . This isomorphism is functorial in S and gives the functor of points in J (X).
jm m
We are mostly interested in the underlying scheme J (X) of the log jet scheme.
m
The main theorem we prove is the following:
Theorem 1.1. Let X be a normal, local complete intersection log variety. Then X
has canonical singularities if and only if J (X) is irreducible for all m > 0.
m
We need to explain the notion of a normal, complete intersection log variety and
when such a variety has canonical singularities. A log variety X is a local complete
intersection log variety if locally it is a complete intersection in a toric variety Y,
with the log structure M being the restriction of the standard log structure M
X Y
on Y. We also require that X intersect the torus orbits of Y in proper dimension.
The log variety X is normal if the underlying variety X is normal.
This work was partially supported by NSERCDiscovery grant.
1
2 KALLEKARUANDANDREWSTAAL
Todefinecanonical singularities, weneedthecanonicaldivisorKX ofalogvariety
X. This is defined more generally below, but for complete intersections in a toric
variety it can bedescribedas follows. Let T ⊂ Y bea toric variety with its standard
log structure. Then KY := KY +DY, where KY is the canonical divisor of Y and
DY is the reduced divisor YrT. The divisor KY is Cartier (in fact, it is trivial). If
X is a normal, complete intersection in Y, we define D = D | and the canonical
X Y X
divisor KX = KX +DX. By adjunction, this divisor is again Cartier.
Now consider a morphism of log varieties f : X˜ → X, such that the underlying
morphism of varieties f : X˜ → X is proper birational. The discrepancy divisor of
f is defined as the difference KX˜ −f∗(KX). The discrepancy divisor is required to
equal D −f∗(D ) away from the exceptional locus. We say that the log variety
X˜ X
X has canonical singularities if the discrepancy divisor is effective for every proper
birational log morphism f.
We will show that the log variety X is canonical if and only if the pair (X,D )
X
is log canonical near D and XrD is canonical. This implies that to determine
X X
if X is canonical, it suffices to consider one resolution f : X˜ → X. Similarly, we
show that the irreducibility of the log jet scheme J (X) can be expressed in terms
m
of irreducibility and dimension of ordinary jet schemes. This reduces the proof of
Theorem 1.1 to the cases considered in [9, 10].
In the theorem we require X to be a normal log variety. In particular, all toric
varieties are assumed to be normal. This is used in the proof to simplify the treat-
ment of Cartier divisors on X. In Kato’s theory of log schemes [5], normality does
not play a large role. For example, log smooth varieties are usually not normal. It
may be possible to extend the theorem to the non-normal case.
TherequirementofthetheoremthatXisalocalcompleteintersectionisessential.
Similarly to the case of ordinary jets, it implies for example that the jet scheme
J (X) has no components of small dimension.
m
Acknowledgements. We thank Mircea Musta¸t˘a for his help with the statement
and proof of the key lemma in Section 4.
2. Log varieties
We work over an algebraically closed field k of characteristic zero. The scheme-
theoretic notation follows [3] and for log schemes we refer to [5]. All sheaves are
taken in the Zariski topology.
2.1. Basics about log varieties. A semi-group is a set with a binary, associative
operation. A monoid is a unitary semi-group; it has a unique identity element.
We only consider commutative monoids, and every homomorphism of monoids is
required to preserve the identity. Let P = (P,+) denote a monoid throughout. Let
X = (X,O )beascheme. Undermultiplication,O formsasheafof(commutative)
X X
monoids. A pre-log structure (pre-log str) on X is the supplemental data of a sheaf
M of (commutative) monoids on X and a homomorphism of sheaves of monoids
X
α : M → O . We usually refer only to “the pre-log str M ,” suppressing α ,
X X X X X
andtoαwherenoambiguity arises. Amorphismofpre-logstructuresisamorphism
of sheaves of monoids M → M′ compatible with the maps to O .
X X X
SINGULARITIES OF LOG VARIETIES VIA JET SCHEMES 3
A pre-log scheme is a scheme endowed with a pre-log str. Pre-log schemes are
denoted in the form X = (X,M ). A pre-log scheme X such that X is a variety is
X
called a pre-log variety.
Let X = (X,M ) and Y = (Y,M ) be pre-log schemes. A morphism (or log
X Y
morphism) from X to Y is a pair of data given by a morphism f : X → Y and a
morphism of pre-log structures f♭ : M → f M . Such a morphism is denoted by
Y ∗ X
(f,f♭) or simply f.
A pre-log str M on X is called a log structure (log str) on X if restricting α
X
induces an isomorphism α−1O∗ →˜O∗ . A log scheme is a scheme endowed with a
X X
log str. If X is both a pre-log variety and a log scheme, we call X a log variety. A
morphism of log schemes is a morphism of the underlying pre-log schemes.
Associated to any pre-log str on X is a natural log str Ma on X formed by the
X
pushout of the homomorphisms α−1O∗ → O∗ and α−1O∗ ֒→ M :
X X X X
Ma ←−−−− M
X X
x x
 
 
O∗ ←−−−− α−1O∗ .
X X
The log str Ma is called the log str associated to M . This log str is universal for
X X
homomorphisms from M to log structures on X.
X
Example 2.1. The trivial log str on X is defined by M = O∗ and the inclusion
X X
morphism α :M → O . This log str is initial in the category of all log structures
X X
on X. For any log scheme Y, a morphism of log schemes Y → X is equivalent to a
morphism of schemes Y → X.
Similarly, the log structure given by M = O and the identity morphism α :
X X
M → O is final in the category of all log structures on X. In this case, for any
X X
log scheme Y, a morphism of log schemes X → Y is equivalent to a morphism of
schemes X → Y.
Let X be any scheme, Y a pre-log scheme, and f : X → Y a morphism of
schemes. Theinverseimagesheaff−1M togetherwiththecompositionf−1M →
Y Y
f−1O → O defines a pre-log str on X. Further, if Y is a log scheme, the inverse
Y X
image log str is defined to be (f−1M )a and is denoted f∗M . The inverse image
Y Y
log str is well-behaved; Kato notes that if Y is a pre-log scheme, then (f−1M )a ≃
Y
f∗Ma (cf. [5]). When f : X → Y is an open or closed embedding, we may denote
Y
f∗M by M | .
Y Y X
On the other hand, if X is a log scheme, Y is any scheme, and f : X → Y is a
morphism of schemes, then the direct image log str is the fibre product of the mor-
phisms O → f O and f M → f O (with the obvious structure morphism),
Y ∗ X ∗ X ∗ X
log
denoted by f M .
∗ X
Example 2.2. Let  : U ֒→ X, be an open embedding, give U the trivial log str
M = O∗ and let M = logM . The local sections of M are all the regular
U U X ∗ U X
functionsthat areinvertible on U.We call thisM thelog str associated to the open
X
embedding U ⊂ X.
4 KALLEKARUANDANDREWSTAAL
In practice we only work with certain well-behaved (pre-)log structures. We will
need some notions about monoids in order to introduce them.
Every monoid P contains its group of units P∗; the quotient monoid P/P∗ is
denoted P. Concretely, P is the set of cosets {p + P∗ : p ∈ P} with the natural
operation.
ToeverymonoidP thereisassociatedagroupPgp,thegroupificationofP,defined
by the obvious universal property for homomorphisms from P to arbitrary abelian
groups. Let p,p′ and n,n′ denote elements in P. The group Pgp is concretely de-
scribed as the quotient of the product (of monoids) P ×P by the equivalence
(p,n) ∼ (p′,n′) ⇐⇒ ∃m∈ P : p+n′+m = p′+n+m.
If every equality p+n = p+n′ in P implies n = n′, then P is called integral.
Alternatively, P is integral if the natural homomorphism P → Pgp taking p 7→ (p,0)
is injective. A finitely generated and integral monoid is called fine. P is called
saturated if for any p∈ Pgp such that mp = p+p+···+p lies in P for some integer
m > 0, we have that p ∈ P.
These notions carry over to sheaves of monoids. Let the subsheaf of units of a
sheaf of monoids M on X be denoted M∗ . We also denote the sheaf of monoids
X X
M /M∗ by M , and the groupification of M by Mgp.
X X X X X
A monoid P and a homomorphism of monoids P → O (X) determines the con-
X
stant sheaf of monoids P on X, the pre-log str α : P → O , and the associated
X X X
logstr Pa → O . Alog structureM isquasi-coherent if atany pointof X thereis
X X X
amonoid P suchthat locally M is isomorphicto Pa.More precisely, for any point
X X
x ∈ X, the pair of data comprised of an open neighbourhood of x, say  : U ֒→ X,
andamonoid P with ahomomorphismP → O (U)together definesachart of M
X X
at x if ∗M ≃ Pa as log structures on U. A quasi-coherent log scheme carries a
X U
quasi-coherent log str, or equivalently, admits a chart at every point. A log scheme
is coherent (resp. integral, fine, saturated) if every point admits a chart (U,P) with
finitely generated (resp. integral, fine, saturated) P. We are mostly interested in fine
andsaturatedlogvarieties. IfthelogstructureM isfineandsaturated, thenevery
X
point x ∈ X has a chart (U,P) such that the natural map induces an isomorphism
P ≃ M (Lemma 1.6 in [6]).
X,x
Example 2.3. Let X be a variety, D ⊂ X a divisor and V = XrD. The log str
on X associated to the open embedding V ⊂ X is fine and saturated. To construct
a chart (U,P) at x ∈ X, let P be the monoid of effective Cartier divisors near
x that are supported on D. This is an integral, finitely generated and saturated
monoid. Choose local equations for elements in a basis of Pgp, defined on an open
neighborhood U of x. Monomials in these local equations will give local equations
of elements of P and hence a morphism of monoids P → O (U).
X
2.2. Themonoid algebra. GivenamonoidP,themonoid algebrak[P]isthesetof
finiteformalsums{ a χp :p ∈ P,a ∈ k} withformaladdition, andmultiplication
P p p
induced by the monoid operation (namely χp ·χp′ = χp+p′). One may consider the
constant sheaf P on the space X = Speck[P]; it yields a natural pre-log str on
X
X arising from the canonical homomorphism P ֒→ k[P]. Clearly the log scheme
X = (Speck[P],Pa) is quasi-coherent, and is integral, fine, or saturated if and only
X
SINGULARITIES OF LOG VARIETIES VIA JET SCHEMES 5
if the monoid P is integral, fine or saturated. We call Pa the standard log str on
X
the scheme Speck[P]. When P is a fine and saturated monoid, then X = Speck[P]
is an affine toric variety (cf. [2]). The standard log str on X is the same as the log
str associated to the open embedding of the big torus T ⊂ X.
Let (X,M ) be a log scheme. To give a chart (U,P) at x ∈ X is the same as to
X
give a morphism of log schemes (U,M | ) → (Y = Speck[P],Pa), such that the
X U Y
pullback of Pa is isomorphic to M | . If the log str M is fine and saturated, we
Y X U X
can choose P to be fine and saturated. Moreover, if X is of finite type, we can make
P biggerifnecessary(byaddingunitsfromO∗)andassumethatU → Speck[P]isa
U
closedembedding. Thus,afineandsaturatedlogvarietyislocallyaclosedsubvariety
of a toric variety, with the log str M being the restriction of the standard log str
X
on the toric variety.
Generalizing the previous argument slightly, let Y = SpecR be an affine scheme
with log str associated to the morphism P → R for a monoid P. If X is any log
scheme, then a morphism of log schemes X → Y is determined by a commutative
diagram
M (X) ←−−−− P
X
 
 
y y
O (X) ←−−−− R.
X
If R = k[P] is the monoid algebra of P and Y has the standard log str, then the
diagram is determined solely by the homomorphism of monoids P → M (X).
X
2.3. Stratification of log varieties. Let X = (X,M ) be a fine and saturated
X
log variety, M the stalk of the sheaf of monoids M at a point x ∈ X, and
X,x X
(M )gp its associated group. This is a finitely generated free abelian group; let
X,x
r(x) be its rank. The function r :X → Z divides X into a disjoint union:
≥0
X =∪ X , X = r−1(i).
i i i
Each locally closed set X can be given the structure of a locally closed subscheme
i
of X as follows. Near a point x ∈ X the ideal of X in O is generated by
i i X
α(M rM×).
X X
Example 2.4. When Y is a toric variety with its standard log str, the stratum
Y is the union of codimension i torus orbits with reduced scheme structure. A
i
chart (U,P) of a fine and saturated log scheme X defines a morphism of schemes
U → Y = Speck[P]. The strata in U are the inverse images of the strata in Y.
Assumption 2.5. We will assume that X has codimension i in X for every i.
i
The codimension of X cannot begreater than i. If the codimension is less than i,
i
but X is non-empty, then the log jet schemes of X are reducible (see Example 3.3
0
below). With some extra work it may be possible to define the canonical divisor
KX for such varieties and prove that they do not have canonical singularities. For
simplicity, we will require the assumption.
6 KALLEKARUANDANDREWSTAAL
2.4. Local complete intersection log varieties. Recall thatafineandsaturated
log variety X = (X,M ) is locally a closed subset of a toric variety Y =Speck[P],
X
such that the log str M is the restriction of the standard log str on Y. We say
X
that X is a local complete intersection log variety if the closed embedding can be
chosen to be a complete intersection. More precisely:
Definition 2.6. A log variety X is a local complete intersection log variety if it
satisfies Assumption 2.5 and every point x ∈ X has a chart (U,P) with P a fineand
saturated monoid such that U → Y =Speck[P] is a closed embedding whose image
is a complete intersection in Y.
Note that a local complete intersection log variety is by definition fine and satu-
rated. Assumption 2.5 means that the closed set U ⊂ Y intersects all torus orbits
in proper dimension.
Example 2.7. Let X be a non-singular variety, D ⊂ X a reduced divisor. Let
the log str M be associated to the open embedding XrD ⊂ X. Then X is
X
a local complete intersection log variety. Near x ∈ X, let f ,...,f define the
1 m
components of D. Then locally X is isomorphic to the graph of f × ··· × f
1 m
in Y = X × Speck[z ,...,z ]. Give Y the log str associated to the embedding
1 m
{z ···z 6= 0} ⊂ Y. Then X is a complete intersection in Y and the log str on X is
1 m
the restriction of the log str on Y. The variety Y is not in general toric. However,
sinceY isnon-singular,locallyY admitsan´etalemorphismtoatoricvarietyZ,with
log str on Y pulled back from Z. This means that locally Y can be embedded as a
hypersurfacein the toric variety Z×k∗ so that the morphism to Z is the projection.
2.5. Log varieties with canonical singularities. We say that a log variety X is
normal if the underlying variety X is normal. Consider normal, fine and saturated
log varieties satisfying Assumption 2.5.
The canonical divisor of a log variety X is
KX = KX +DX,
where K is the canonical divisor of the variety X and D is an effective Weil
X X
divisor defined as follows. Let x ∈ X be the generic point of an irreducible divisor
D. If the ideal generated by α(M rM×) vanishes to order a at x, then D appears
X X
with coefficient a in D .
X
Since X satisfies Assumption 2.5, the support of the divisor D is the closure of
X
the stratum X . The coefficient of D in D is equal to the length of the scheme X
1 X 1
at the generic point of D.
If the log str M is associated to an open embedding U ⊂ X, then D = XrU
X X
is a reduced divisor. In particular, if Y is a toric variety with its standard log str,
then DY = YrT and hence KY = 0. From the adjunction formula we get that KX
is a Cartier divisor for any local complete intersection log variety X.
Amorphismoflogvarieties f :Y → Xiscalled proper birational iftheunderlying
morphism of varieties f : Y → X is proper birational. Assume that KX is Cartier,
f proper birational, and let
(2.1) KY = f∗KX+XaiEi,
SINGULARITIES OF LOG VARIETIES VIA JET SCHEMES 7
whereE areirreducibledivisorson Y; to make thecoefficients a unique, werequire
i i
that if E is notexceptional, thena is the coefficient of E inD −f∗(D ). Thelog
i i i Y X
variety X has canonical singularities (or simply, is canonical) if thetotal discrepancy
divisor a E is effective for every proper birational f :Y → X.
P i i
Let us recall singularities of pairs (cf. [8]). A pair (X,D) consists of a normal
variety X andadivisorD inX,suchthatK +D isCartier. Foraproperbirational
X
morphism Y → X, let
(2.2) KY = f∗(KX +D)+XbiEi,
where E are irreducible divisors on Y. One assumes that if E is not exceptional,
i i
then b is the coefficient of E in −f∗(D). The divisor b E is called the total
i i P i i
discrepancy divisor of f. The pair (X,D) is log canonical (resp. canonical) if the
total discrepancy divisor has coefficients b ≥ −1 (resp. b ≥ 0). When D = 0, then
i i
the variety X is canonical if b ≥ 0 for all i.
i
Proposition 2.8. A log variety X is canonical if and only if the pair (X,D ) is
X
log canonical and XrD is canonical.
X
Proof. Let Y = (Y,M ) be a log variety. Define Y′ = (Y,M′ ), where M′ is
Y Y Y
the log str associated to the open embedding YrD ⊂ Y. Then the image of
Y
α :MY → OY lies in the submonoid MY′ ⊂ OY, hence the identity morphism of Y
extends uniquely to a log morphism Y′ → Y. If f : Y → X is a proper birational
morphism, we can compose this with Y′ → Y. After such a composition, the
discrepancy divisor will not increase. Thus, to check if X is canonical, it suffices to
consider proper birational morphisms from Y, where the log str M is associated
Y
to an open embedding U ⊂ Y. By the same argument, it suffices to take U =
Yrf−1(D ). In this case D = f−1(D ) with reduced structure. Now comparing
X Y X
thediscrepancy divisors for X and(X,D ) as in theformulas (2.1)and (2.2)above,
X
we see that a = b +1 if E maps to D , and a = b otherwise. This implies the
i i i X i i
statement of the proposition. (cid:3)
As in the case of ordinary varieties, having canonical singularities can be checked
on one resolution. We say that f :(Y,D ) → (X,D ) is a log resolution of the pair
Y X
(X,D ) if Y is non-singular, f−1(D ) is a divisor, D =f−1(D )red is the divisor
X X Y X
with reduced structure, and D together with the exceptional locus of f is a divisor
Y
of simple normal crossings.
Corollary 2.9. Let X be a log variety such that KX is Cartier, and f : (Y,DY) →
(X,D ) a log resolution. Then X has canonical singularities if and only if the
X
divisor
K +D −f∗(K +D )
Y Y X X
is effective. (cid:3)
3. Log jet schemes
Let j be the scheme j = Speck[t]/(tm+1) with its trivial log str. For any log
m m
scheme S the product S×j exists in the category of schemes with log structures,
m
as described in [5]. Concretely, since j is a one point space, S and S × j are
m m
homeomorphic. Since j has the trivial log str, the log str on S ×j is the log str
m m
8 KALLEKARUANDANDREWSTAAL
associated to the pre-log str α : M → O . This is defined by the pushout
S S S×jm
diagram:
M ←−−−− M
S×jm S
x x
 
 
O∗ ←−−−− α−1O∗ .
S×jm S S×jm
Note that α−1O∗ = O∗, and
S S×jm S
O∗ = O∗ ×G,
S×jm S
where G is the sheaf of multiplicative groups
G = {1+a t+···a tm :a ∈ O }.
1 m i S
Itfollows that M is the productof sheaves M ×G, withα : M ×G → O
S×jm S S S×jm
mapping
α : (m,1+a t+···a tm) 7→ α (m)·(1+a t+···a tm).
1 m S 1 m
Let X be a fine and saturated log scheme. The mth log jet scheme of X, which
we denote by Jm(X) = (Jm(X),MJm(X)), represents the functor
S 7→ Hom (S×j ,X)
log.sch m
from the category of log schemes to that of sets. The existence of log jet schemes is
proved in [1].
It is well-known that if X ⊂ An is a closed subscheme defined by l equations,
then the jet scheme J (X) ⊂ J (An) ≃ A(m+1)n is a closed subscheme defined by
m m
(m+1)l equations. In the next subsections we first recall the local equations for the
ordinary jet scheme and then generalize this to the case of log jet schemes.
3.1. Equationsdefiningtheordinaryjetscheme. LetY = An = Speck[x ,...,x ].
1 n
Then
(j)
J (Y) = Speck[x ] .
m i i=1,...,n;j=0,...,m
(j) (j)
Let S bea scheme, and R = O (S). AmorphismS → J (Y) given by x 7→ r ∈
S m i i
R corresponds to the S-valued jet S ×j → Y given by
m
tj
(j)
(3.3) xi 7→ Xri j!.
j
(j)
The coordinate ring of J (Y) has a k-linear differential operator d : k[x ] →
m i
(j) (j) (j+1) (m)
k[x ] that maps x 7→ x , x 7→ 0, and satisfies the Leibniz rule. If
i i i i
f(x ,...,x )∈ k[x ,...,x ], then
1 n 1 n
tj tj tj
(3.4) f(Xx(1j)j!,...,Xx(nj)j!) = Xdjfj!.
j j j
It follows from this that if X ⊂ Y is the closed subscheme X = V(f ,...,f ), then
1 l
J (X) ⊂ J (Y) is also a closed subscheme defined by
m m
J (X) = V(djf ) .
m i i=1,...,l;j=0,...,m
SINGULARITIES OF LOG VARIETIES VIA JET SCHEMES 9
Forlaterusewementionthattheoperatordextendstotheringk[x±1,...,x±1][x(j)]
1 n i
and formula (3.4) also holds for Laurent polynomials f ∈ k[x±1,...,x±1].
1 n
3.2. Equations defining the log jet scheme. Now let P ⊂ Pgp ≃ Zn be a fine,
saturated monoid, and Y = Speck[P] a toric variety with its standard log str. Let
e ,...,e ∈ P be a basis for Pgp, and let x ,...,x ⊂ k[P] be the corresponding
1 n 1 n
monomials. Note that elements of k[P] are then Laurent polynomials in x ,...,x .
1 n
We claim that the log jet scheme of Y is
(j)
x
J (Y) = Speck[P][ i ] ,
m i=1,...,n;j=1,...,m
x
i
with log str induced by P ֒→ k[P]. A morphism of log schemes S → J (Y) is a
m
commutative diagram
M (S) ←−−−− P
S
 
 
y y
x(j)
O (S) ←−−−− k[P][ i ].
S xi
x(j)
It is determined by a monoid homomorphism P → M (S) and elements i 7→
S xi
r(j) x(j) r(j)
i ∈ R = O (S). (Note that we view i as variables and i as some elements
ri S xi ri
in R, not necessarily quotients of two elements.)
An S-valued jet in Y is a morphism S×j → Y, which corresponds to a com-
m
mutative diagram
M (S ×j ) ←−−−− P
S×jm m
 
 
y y
R[t]/(tm+1) ←−−−− k[P].
This is determined by a homomorphism of monoids P → M (S × j ). The
S×jm m
monoid M (S ×j ) is the product of the monoid M (S) with the group
S×jm m S
G = {1+a t+···a tm|a ∈ R} ⊂(R[t]/(tm+1))∗.
1 m i
HencethediagramaboveisdeterminedbyahomomorphismofmonoidsP → M (S)
S
and a homomorphism of groups Pgp → G. Let the group homomorphism be given
by
r(j)tj
ei 7→ 1+X i .
r j!
i
j
This gives a bijection between S-valued points of the log jet scheme and S-valued
jets in Y. One can easily check that this is an isomorphism of functors.
x(j)
Suppose we have a morphism S → J (Y), given by φ : P → M (S) and i 7→
m S xi
r(j) (j) r(j)
i ∈ R. Set r = α(φ(e )) ∈ R, and r = r · i . Then the ordinary S-valued jet
ri i i i i ri
into Y underlying the S-valued jet into Y is given by a k-algebra homomorphism
10 KALLEKARUANDANDREWSTAAL
k[P] → R[t]/(tm+1). It maps x ∈ P to
i
r(j)tj tj
xi 7→ ri(1+X ri j!)= ri +Xri(j)j!.
i
j>0 j>0
This formula can be compared with (3.3) and it is the reason for denoting the
x(j)
coordinates on the log jet scheme by i .
xi
Later, in the proof of the main theorem, we will need to consider birational
morphisms of toric varieties and the induced morphisms of log jet schemes. Let
P ⊂ Q be two fine and saturated monoids, such that Pgp = Qgp. The inclusion of
monoids gives a morphism of toric varieties f : Y′ = Speck[Q] → Y = Speck[P].
This map is birational. With notation as above, we choose a basis e ,...,e ∈ P for
1 n
Pgp and denote by x ,...,x the corresponding monomials in both k[P] and k[Q].
1 n
Then
(j) (j)
x x
J (Y′) = Speck[Q][ i ], J (Y) = Speck[P][ i ],
m m
x x
i i
andoneeasilychecksthattheinducedmorphismoflogjetschemesJ (Y′)→ J (Y)
m m
x(j) x(j)
takes i to i . This result can be restated by saying that the diagram
xi xi
J (Y′) −−−−→ J (Y)
m m
 
 
y y
Y′ −−−−→ Y
is a fibre square. In fact, the diagram is also a fibre square in the category of log
schemes when we give each scheme its log structure.
To study the log jet schemes of closed subschemes of Y, we need to define the
x(j) x(j)
k-linear derivation d : k[P][ i ] → k[P][ i ]. Note that we have an inclusion of
xi xi
rings
(j)
x
k[P][ i ] ⊂ k[x±1,...,x±1][x(j)].
x 1 n i
i
(j)
The derivation d on k[x ,...,x ][x ] defined in the case of ordinary jet schemes
1 n i
extends to a derivation on the ring k[x±1,...,x±1][x(j)]. The subring k[P][x(ij)] is
1 n i xi
invariant by this extension, so we define the desired derivation d as the restriction
of the extended d. Invariance of the subring under d follows from:
(1)
x
d(Yxaii)= (Yxaii)(Xai xi ),
i
i i i
(j) (j+1) (1) (j) (j+1) (1) (j)
x x x −x x x x x
d( i )= i i i i = i − i i .
x x2 x x x
i i i i i
Now let X ⊂ Y be a closed subscheme defined by X = V(f ,...,f ). Let the log
1 l
str on X be the restriction of the log str on Y.