This paper introduces the concept of successive minima for line bundles on algebraic varieties, linking geometric invariants like volume, width, and Seshadri constants, and relates them to toric geometry.
Contribution
It defines and studies successive minima of line bundles, connecting them to classical invariants and extending their understanding in algebraic and toric geometry.
Findings
01
Successive minima relate to width and Seshadri constants.
02
Volume equals the product of successive minima.
03
In toric varieties, minima correspond to classical lattice minima.
Abstract
We introduce and study the successive minima of line bundles on proper algebraic varieties. The first (resp. last) minima are the width (resp. Seshadri constant) of the line bundle at very general points. The volume of the line bundle is equivalent to the product of the successive minima. For line bundles on toric varieties, the successive minima are equivalent to the (reciprocal of) usual successive minima of the difference of the moment polytope.
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TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Algebraic structures and combinatorial models
Full text
Successive minima of line bundles
Florin Ambro
Institute of Mathematics “Simion Stoilow” of the Romanian Academy
We introduce and study the successive minima of line bundles on proper algebraic varieties.
The first (resp. last) minima are the width (resp. Seshadri constant) of the line bundle
at very general points. The volume of the line bundle is equivalent to
the product of the successive minima. For line bundles on toric varieties, the
successive minima are equivalent to the (reciprocal of) usual successive minima
of the difference of the moment polytope.
The motivation for this paper is the classical problem in Algebraic Geometry of finding numerical criteria for a linear system on an
algebraic variety to be non-empty, or to define a birational embedding. The equivalent problem in Geometry of Numbers
is finding numerical criteria for a convex body to contain sufficiently many lattice points.
For a complex projective manifold X, Demailly [5] introduced the Seshadri constant ϵ(L,x)
of an ample line bundle L at a point x of X. This invariant is constant if x is very general, denoted
ϵ(L) and called maximal Seshadri constant, and one can show that if ϵ(L) is sufficiently large,
then the linear system ∣KX+L∣ is non-empty, and even defines a birational embedding (see [5, 7], or Theorem 5.1).
For a convex body □⊂Rd, the flatness theorem of Khinchin states that if the lattice width w
of □ is sufficiently large, then □ contains a lattice point (see [12]). The starting point of
this paper is to observe that the flatness theorem is just a special (toric) case of Demailly’s generation for adjoint line bundles
in terms of maximal Seshadri constants. The key point is that on toric varieties, the maximal Seshadri constant
is equivalent to the lattice width of the moment polytope:
Theorem 0.1**.**
Let X=TNemb(Δ) be a proper toric variety, of dimension d. Let L be an invariant Cartier divisor on X,
with Seshadri constant ϵ at a general point of X. Let □L⊂MR be the moment polytope,
let w be the lattice width of □L. Then
[TABLE]
The equivalence between w and ϵ not only reproves the flatness theorem, but generalizes
it as follows:
Theorem 0.2**.**
Let □⊂Rd be a compact convex set, of lattice width w.
If w>d2+d, there exist m0,…,md∈Zd∩int(□)
such that m1−m0,…,md−m0 form a basis of Rd.
The proof of the equivalence involves the transference theorem of Mahler, and the fact that w is
the first minimum of Minkowski for the polar body (□L−□L)∗⊂NR. Since ϵ
is proportional to the first minimum of Minkowski, we may think of 1/ϵ as a d-th
successive minimum of □L−□L⊂MR via the transference theorem. The question arises if the other successive minima of this [math]-symmetric body
have a meaning on algebraic varieties which are not necessarily toric, and if they have a connection with Seshadri constants.
The answer turns out to be positive! We introduce in this paper a sequence of successive minima for a line bundle on an algebraic
variety, such that the last minimum coincides with the classical Seshadri constant, and in the toric case, these minima are
equivalent to the (reciprocal of) successive minima of the [math]-symmetric body associated to the moment polytope.
Let X be a proper complex variety, let L be a Cartier divisor on X.
Let x∈X be a point. For a real number t≥0, we define Bs∣Ixt+L∣Q to be the common zero
locus of sections s∈Γ(nL)(n≥1) such that s vanishes at x of order strictly larger than nt.
Then Bs∣Ixt+L∣Q is a closed subset of X, increasing with respect to t, and equal to X for t sufficiently large.
For integers i≥1, the i-th successive minimum of L at x is defined as
[TABLE]
We obtain a sequence ϵ1(L,x)≥ϵ2(L,x)≥⋯≥ϵd(L,x)≥0=ϵd+1(L,x),
where d=dimX.
As we will see,
the invariant ϵi(L,x) does not depend on a very general point x, is denoted ϵi(L),
and called the i-th minimum of L at a very general point. If κ(L)≤0, then ϵ1(L)=0. If
L has Iitaka dimension κ≥1, it turns outs that ϵκ(L)>0=ϵκ+1(L).
We can show that ϵi(L) are numerical invariants in case L is big.
If L is semiample and x∈X is a smooth point, ϵd(L,x) coincides with the Seshadri constant
ϵ(L,x) introduced by Demailly [5]. In particular, if L is semiample, ϵd(L) coincides with the
maximal Seshadri constant of L, usually denoted ϵ(L).
The invariant ϵ1(L,x) is the largest asymptotic multiplicity that can be imposed at x
on sections of multiplies of L. It appears in the work of Nakamaye [17]. If x∈X is a smooth point, it coincides with the
width of L at the geometric valuation induced by the exceptional divisor on the blow-up at x (see [1]).
Due to this property, we call ϵ1(L,x) the width of L at x.
We also relate the volume of L with the successive minima. By a classical argument for counting jets,
we have inequalities
[TABLE]
The second main result of the paper is the analogue of Minkowski’s second main theorem,
namely the volume of L is equivalent to the product of successive minima of L at very general points:
Theorem 0.3**.**
Let X be a proper complex variety of dimension d, let L be a Cartier divisor on X. Let vol(L) be the volume
of L, let ϵi(L) be the successive minima of L at very general points. Then
[TABLE]
In particular, for a big divisor L on X we have inequalities
[TABLE]
Understanding when the ratio ϵ1(L)/ϵd(L) is too large is an interesting problem (see Nakamaye [17]).
Finally, we show that for line bundles on toric varieties, our successive minima are equivalent to the
reciprocal of usual ones:
Theorem 0.4**.**
Let X=TNemb(Δ) be a proper toric variety, of dimension d. Let L be a big invariant Cartier divisor on X,
with successive minima ϵi at a very general point. Let □L⊂MR be the moment polytope,
let λ1,…,λd be the successive minima of the [math]-symmetric convex body □L−□L⊂MR,
let λ1∗,…,λd∗ be the successive minima of the polar body (□L−□L)∗⊂NR.
Then ϵi,λi−1,λd−i+1∗ are all equivalent. More precisely,
[TABLE]
Our definition of successive minima is inspired by the equivalent definition of Seshadri constants due to Eckl [6, Theorem 1.1],
and by the methods developed by Ein, Küchle, Lazarsfeld [7] and Nakamaye [16, 17, 18],
while studying effective lower bounds for maximal Seshadri constants. Theorem 0.3, in the toric case, is
just a restatement of Minkowski’s second main theorem. It is implicit in the work of
Nakamaye when X is a surface [17, Proof of Corollary 3],
and our proof is similar to his, except that we introduce certain convex polytopes in
his method of counting of jets. Eventually, Theorem 0.3 reduces to a simple postulation problem in the projective
space (Proposition 1.7).
A technical improvement in the study of Seshadri constants is that we no longer require positivity of the line bundles, or
that the ambient space is normal.
On the other hand, our invariants may not be numerical if the line bundle is not big. We hope that successive minima,
and other tools from the Geometry of Numbers, may be useful in the study of adjoint linear systems.
We outline the content of this paper. In Section 1 we setup the notation, and recall basic facts about multiplicities,
and linear systems. We also prove an inequality for symbolic powers of ideals in the projective space,
which is elementary, but new to us. In Section 2, we define successive minima for a line
bundle, study some basic properties, and compare it with Seshadri constants. In Section 3 we prove that the volume
is equivalent to the product of the successive minima. In Section 4, we estimate the successive minima for line
bundles on toric varieties. In Section 5, we generalize the flatness theorem of Khinchin, as an application of known
results on adjoint linear systems, plus the estimates in Section 4. In Section 6 we recall the statements of the second
main theorem of Minkowski, and the transference theorem of Mahler. We also present a proof of the transference theorem
in dimension two, with sharp bounds.
**Acknowledgments **.
The first author is grateful to Max Planck Institut für Mathematik in Bonn for its hospitality and financial support.
The second author was supported by Grant-in-Aid for Scientific Research 17K14162. We would like to thank the referee
for useful comments and suggestions.
1. Preliminary
Throughout this paper, an algebraic varietyX is a scheme of finite type,
reduced and irreducible, defined over an algebraically closed field k, of
characteristic zero (the assumption chark=0 is used in Lemma 2.18 and
its consequences in Section 3, and in Theorem 5.1).
Vanishing orders
Let X be an algebraic variety, L an invertible OX-module, and s∈Γ(X,L)
a global section. The order of s at a closed pointx∈X, denoted ordx(s), is the supremum after
all integers n≥0 such that sx∈Ixn⋅Lx. The order is
+∞ if s=0, and a non-negative integer if s=0.
If f:X′→X is a morphism of algebraic varieties and f(x′)=x,
then f∗s∈Γ(X′,f∗L) and ordx′(f∗s)≥ordx(s).
In particular, if x∈X′⊆X is a closed subset, then ordx(s)≤ordx(s∣X′).
Suppose X is smooth at the general point of a subvariety Z⊆X. The order of s at Z
is defined as the order of s at a general point of Z.
Order bounds
For an effective Cartier divisor D on X, the order of D at x, denoted ordx(D), is defined
as the order at x of a local equation of D.
Lemma 1.1**.**
Let X be a projective algebraic variety, of dimension d,
let A be a very ample divisor, and D an effective
Cartier divisor on X. Then ordx(D)≤(D⋅Ad−1) for every closed point x∈X.
Proof.
Since A is very ample, there exist H1,…,Hd−1∈∣A∣ such that
C=∩i=1d−1Hi is an effective cycle passing through x, none of its components being contained in D.
Then
[TABLE]
We note that the first inequality follows from [10, page 233] even if X is singular.
∎
Volume
Let X be a proper algebraic variety, of dimension d. Let L be a Cartier divisor on X,
let R(L)=⊕n≥0Γ(X,OX(nL)) be the induced N-graded ring. Let
R=⊕n≥0Rn⊆R(L) be a graded subalgebra. The volume of R
is defined as
[TABLE]
The limsup is in fact a limit over sufficiently divisible n.
The volume of L, denoted vol(L), is defined as the volume of R(L).
We usually denote Γ(X,OX(nL)) by Γ(nL).
Iitaka map
One may define the Iitaka dimension and Iitaka map for divisors on a variety which may not
be normal. These are birational invariants if the ambient space is normal.
Let X be a proper variety, let L be a Cartier divisor on X. For n≥1 such that Γ(nL)=0,
let ∣nL∣ be the induced linear system on X. A basis s0,…,sN of Γ(nL) induces
a rational map ϕ∣nL∣:X⇢∣nL∣, with image Xn.
Define the Iitaka dimensionκ(L) of L to be the maximum of dimXn after all such n,
and −∞ if Γ(nL)=0 for all n≥1.
Suppose κ(L)≥0. For each n≥1 such that Γ(nL)=0, let Qn⊆k(X)
be the field over k generated by {ss′;s′,s∈Γ(nL)∖0}. The dominant rational map
ϕ∣nL∣:X⇢Xn induces an isomorphism k(Xn)→∼Qn. The union
∪∣nL∣=∅Qn is a subfield Q of k(X), since Qn∪Qn′⊆Qn+n′ if
∣nL∣ and ∣n′L∣ are non-empty. Since Q is finitely generated over k, there exists an integer m such
that Qm=Q. For every n≥1, we have Qm=Qnm, that is the rational maps
ϕ∣mL∣:X⇢Xm and
ϕ∣nmL∣:X⇢Xnm differ by a birational map Xnm⇢Xm.
We call ϕ∣mL∣:X⇢Xm the Iitaka map of L. The dimension of Xm
is κ(L).
Suppose κ(L)≥0 and X is normal. The normalization of the graph of the Iitaka map
ϕ∣mL∣:X⇢Xm induces a diagram
[TABLE]
such that X′ is proper and normal, OX=μ∗OX′, and
κ(μ∗L∣Xy′)=0 for a very general point y∈Xm. Note that Xy′ is connected.
Suppose L is semiample, that is the linear system ∣nL∣ has no base points for some
n≥1. Then the Iitaka map ϕ:X→Y is regular, OY=ϕ∗OX, and
nL∼ϕ∗A for some integer n≥1 and a very ample divisor A on Y.
Suppose μ:X′→X is a proper birational morphism. If X and X′ are normal, it follows
that OX=μ∗OX′. In particular, Γ(nL)=Γ(nμ∗L) for every n≥1.
Therefore the rational maps induced by non-empty linear systems ∣nL∣ and ∣nμ∗L∣ differ by
μ. It follows that κ(L)=κ(μ∗L).
A morphism of algebraic varieties f:X→Y is called a contraction if
OY=f∗OX.
Isolated base points of linear systems
Let X be a proper algebraic variety, let L be a Cartier divisor on X such that Γ(L)=0.
A basis s0,…,sN of Γ(L)
induces a rational map ϕ:X⇢PN=∣L∣. Let U=X∖Bs∣L∣,
so that ϕ is regular on U. Then Bs∣Ix(L)∣∩U=ϕ−1ϕ(x)∩U for every x∈U.
Lemma 1.2**.**
Let f:X→Y be a proper contraction of algebraic varieties, let x∈X be a closed point such that
dimxXf(x)=0, where Xf(x)=f−1f(x). Then f is an isomorphism over a neighborhood of f(x).
Proof.
By the semicontinuity of the local dimension of the fibers of f,
there exists an open subset U′∋x such that dimx′Xf(x′)=0 for all x′∈U′. Let
V=Y∖f(X∖U′). Since Xf(x) is [math]-dimensional and connected, it coincides
with {x}. Therefore f(x)∈V. Let U=f−1(V). Then x∈U⊆U′. Therefore
f∣U:U→V is a proper contraction with finite fibers, hence a finite contraction, hence an isomorphism.
∎
Let X be an algebraic variety. Let I⊆OX be an ideal sheaf,
let f:Y→X be the blow-up of X along I. Let E be the exceptional
divisor on Y, defined by I⋅OY=OY(−E). Then the natural homomorphism
In→f∗OY(−nE)
is an isomorphism for every n≥c(I), where c(I)≥0 is a constant depending only on I.
Moreover, c(I)=0 if I is the ideal of a smooth point of X.
Proof.
The statement is local, so we may suppose X=SpecR and Y=ProjS, where S
is the R-graded ring ⊕n≥0In, for some ideal I⊆R. Then
OY(−nE)=OY(n)=S(n)~. The natural homomorphism
S→⊕n≥0Γ(Y,S(n)~)
is an isomorphism in degrees n≥c(I). This gives the claim.
If I is the ideal of a smooth point of X, then I is generated by a regular sequence.
One checks then that c(I)=0.
∎
Let X be a proper algebraic variety, let L be a Cartier divisor on X and x∈X a closed point
such that Bs∣L∣∩U⊆{x} for some open neighborhood U of x in X.
Then x∈/Bs∣nL∣ for n≫0.
Proof.
Let Z be the base locus subscheme of ∣L∣ outside x (possibly empty).
Let f:Y→X be the blow-up of X along IZ. Let IZ⋅OY=OY(−E), so
that E is the exceptional divisor.
Since f is an isomorphism over a neighborhood of x, we may identify x with a point of Y.
A basis of Γ(X,IZ(L))=Γ(X,OX(L)) induces sections of Γ(Y,OY(f∗L−E))
whose common zero locus is away from E. Therefore Bs∣f∗L−E∣⊆{x}. By [9, Theorem 1.10],
x∈/Bs∣n(f∗L−E)∣ for n≥n0. By Lemma 1.3, we may identify
Γ(X,IZn(nL))=Γ(Y,OY(nf∗L−nE)) for every n≥c(IZ).
Therefore x∈/Bs∣IZn(nL)∣ for n≥max(n0,c(IZ)).
Since x∈/Z, we obtain x∈/Bs∣nL∣ for n≫0.
∎
Lemma 1.5** (Ample reduction).**
Let X be proper, L a Cartier divisor and x∈X a closed point such that Bs∣Ix(L)∣={x} near x.
Let Z be the base locus subscheme of ∣L∣, away from x (possibly empty). Let f:X′→X
be the blow-up of X along IZ, with exceptional divisor E. Then there exists a proper
contraction g:X′→Y to a projective algebraic variety Y, such that the following properties hold:
a)
f∗L−E∼g∗A* for some ample Cartier divisor A on Y, and*
b)
f* is an isomorphism over a neighborhood of x, and g is an isomorphism over
a neighborhood of g(x).*
[TABLE]
We obtain natural homomorphisms
R(L)⊇⊕n≥0Γ(IZn(nL))→αR(f∗L−E)≃R(A),
and α is an isomorphism in sufficiently large degrees.
Proof.
By construction, ∣f∗L−E∣ is base point free near E. Therefore
Bs∣f∗L−E∣⊆{x}. By the proof of Lemma 1.4, ∣n(f∗L−E)∣ is base point free
for n≫0. The Iitaka map induces the contraction g, with property a).
For b), it is clear that f is an isomorphism over a neighborhood of x. We identify
x with a point of X′. Since Bs∣Ix(L)∣={x} near x, we obtain
Bs∣Ix(f∗L−E)∣={x} near x. Therefore Bs∣Ix(nf∗L−nE)∣={x} near x,
for all n≥1. Therefore g−1g(x)={x} near x. By Lemma 1.2, g is an
isomorphism over a neighborhood of g(x). Finally, αn is an isomorphism
for n≥c(IZ).
∎
Generation of jets
Let X be an algebraic variety, L a Cartier divisor on X, x∈X a closed point, and p≥0 an integer.
We say that L* generates p-jets at x* if Γ(L)→Ox/Ixp+1 is surjective.
By induction and the snake lemma, this is equivalent to the following property:
the OX-module Ixn(L) is generated by global sections at x, for every integer 0≤n≤p.
Lemma 1.6**.**
Let f:Y→X be the blow-up at x, with exceptional divisor E. Suppose
Ixp+1→∼f∗(IEp+1) and R1f∗(IEp+1)=0. If
Γ(f∗L)→Γ(O(p+1)E) is surjective, then L generates p-jets at x.
The converse holds if the natural homomorphism H1(L)→H1(f∗OY(L)) is injective.
Proof.
Consider the commutative diagram
[TABLE]
The bottom row is exact. The top is also exact, since R1f∗(IEp+1)=0. The first vertical arrow
is an isomorphism, by assumption. The second vertical arrow is injective. Therefore the third
arrow is injective as well. We deduce that the second and third vertical arrows have isomorphic
cokernels, denoted by C. Ignoring the first vertical arrow, tensoring with OX(L)
and passing to global sections on X, we obtain a commutative diagram
[TABLE]
with exact rows. If b is surjective, then a is surjective. The converse holds if c is surjective.
From the exact sequence
[TABLE]
we see that c is surjective if and only if H1(L)→H1(f∗OY(L)) is injective.
∎
If x is a smooth point, then Ixp+1→∼f∗(IEp+1) and R1f∗(Ixp+1)=0 for all p≥0,
and OX=f∗OY. Therefore L generates p-jets at x if and only if
Γ(f∗L)→Γ(O(p+1)E) is surjective.
Postulation
For an integer p≥0, denote Nd(p)={(α1,…,αd)∈Nd;∑i=1dαi=p}.
For a finite set A, denote by ∣A∣ its cardinality.
We will need the following property on the symbolic powers of ideals in the projective space.
Proposition 1.7**.**
Let Zi⊂Pd be an irreducible subvariety of codimension i, for 1≤i≤d.
Let p1,…,pd,q≥0 be integers. Then
[TABLE]
When Z1⊃Z2⊃⋯⊃Zd is a linear flag in Pd, the equality holds.
Proof.
Suppose Z1⊃Z2⊃⋯⊃Zd is a linear flag in Pd.
Choose coordinates [z0:⋯:zd] on Pd such that IZi=(z0,…,zi−1) for all i.
Then Γ(Pd,∩i=1dIZi(pi)⊗OPd(q)) is the k-vector
space with monomial basis z0α0⋯zdαd, where
α∈Nd+1(q) such that α0+⋯+αi−1≥pi for all i.
Equivalently, αi+⋯+αd≤q−pi for all i.
Therefore the inequality becomes an equality if Z∙ is a linear flag.
If d=1, the equality holds from 1). Suppose d>1, and assume by induction that the inequality
holds in smaller dimension. Let H:{λ=0} be a general hyperplane with respect to
Z∙. In particular, H does not contain the point Zd. The short exact sequence
[TABLE]
induces an exact sequence
[TABLE]
Indeed, Γ(Pd,∩i=1dIZi(pi)⊗OPd(q)) consists of homogenous
polynomials F of degree q such that ordx(F)≥pi for every x∈Zi, for all i.
Then ordx(F∣H)≥pi for every x∈Zi∩H. If F∣H=0, then F=λF′ for some
homogeneous polynomial F′ of degree q−1. For x∈Zi∖H we have ordx(F)=ordx(F′).
Then F′ vanishes to order at least pi at a general point of Zi, hence everywhere on Zi.
Moreover, the above exact sequence extends to a short exact sequence if Z∙ is a linear flag:
[TABLE]
Indeed, we may choose coordinates as in 1), and suppose λ=zd. The
part αd≥1 of the set
[TABLE]
corresponds to a monomial basis of
Γ(Pd,∩i=1dIZi(pi)⊗OPd(q−1)), and the part αd=0
corresponds to a monomial basis of Γ(H,∩i=1d−1IZi∩H(pi)⊗OH(q)).
Since the dimensions add up, the sequence is exact to the right as well.
Denote f(q)=h0(Pd,∩i=1dIZi(pi)⊗OPd(q)).
For each 1≤i≤d−1, choose an irreducible subvariety Wi⊆Zi∩H, of codimension i in H≃Pd−1.
The exact sequence gives
[TABLE]
Denote g(q)=h0(Pd,∩i=1dILi(pi)⊗OPd(q)), where L∙ is a
linear flag in Pd. The exact sequence gives
[TABLE]
The inequality in dimension d−1 gives
[TABLE]
We deduce
[TABLE]
We have Γ(Pd,IZd(pd)(q))=0 for q<pd. Therefore f(q)=g(q)=0 for q<pd.
By increasing induction on q≥pd, the above inequality gives f(q)≤g(q) for all q.
∎
For real numbers t1,…,td, define a compact convex set in Rd by
[TABLE]
Note that □(t1,…,td) is not empty if and only if t1,…,td≥0.
If t1≥⋯≥td≥0, we may compute
vol□(t1)=t1, 2vol□(t1,t2)=2t1t2−t22, and
6vol□(t1,t2,t3)=6t1t2t3−3t32t1−3t3t22+t33.
Lemma 1.8**.**
Suppose ti≥0 for all i. Then vol□(t1,…,td)≤∏i=1dti.
Proof.
We have an inclusion □(t1,…,td)⊂[0,t1]×□(t2,…,td). Therefore
[TABLE]
By induction on d, the desired inequality holds.
∎
Using this convex set, we may restate Proposition 1.7 as follows:
[TABLE]
2. Successive minima of line bundles
Let X be a proper algebraic variety, of dimension d. Let L be a Cartier divisor,
with induced graded ring R=⊕n≥0Γ(nL). Let x∈X be a closed point.
For a real number t≥0, denote
[TABLE]
It is a closed subset of X. If t≤t′, then Bs∣Ixt+L∣Q⊆Bs∣Ixt′+L∣Q.
One can rewrite Bs∣Ixt+L∣Q as the intersection of the base locus Bs∣Ixp(qL)∣,
after all integers p,q≥1 such that p>qt. Since
[TABLE]
is a graded ring and X is Noetherian, there exists r≥1 such that
Bs∣Ixt+L∣Q=Bs∣Ix⌊rt⌋+1(rL)∣.
Example 2.1**.**
Suppose L is semiample. Let ϕ:X→Y be the Iitaka map
of L. Then Bs∣Ix0+L∣Q=ϕ−1ϕ(x). It is connected,
since ϕ is a contraction.
We have rL∼f∗A for some r≥1 and A ample on Y.
Then for every t≥0, we have Bs∣Ixt+L∣Q⊆f−1(Bs∣If(x)rt+A∣Q),
and equality holds if f is smooth at x.
Lemma 2.2**.**
Given u≥0, there exists ϵ(u)>0 such that Bs∣Ixt+L∣Q is constant
for t∈[u,u+ϵ(u)).
Proof.
Denote Bt=Bs∣Ixt+L∣Q.
We have Bu=∩l≥1Bu+l1. By monotonicity, and since X is Noetherian,
there exists l≥1 such that Bu=Bu+l1.
∎
Lemma 2.3**.**
There exists a constant c, depending only on X and L, such that
Bs∣Ixt+L∣Q=X for every x∈X and t≥c.
Proof.
By Chow’s Lemma and Hironaka’s
resolution of singularities, there exists a proper birational morphism μ:X′→X such that
X′ is smooth and projective. Let A′ be very ample on X′. Let n≥1 and 0=s∈Γ(X,OX(nL)).
Choose a point x′∈μ−1(x). Then 0=μ∗s∈Γ(X′,OX′(nμ∗L)) and
[TABLE]
where the last inequality follows from Lemma 1.1.
Thus Bs∣Ixt+L∣Q=X for t≥(μ∗L⋅A′d−1).
∎
Recall that the codimension at x of a closed subset x∈Y⊆X is the smallest
codimension of an irreducible component of Y passing through x. Since X is irreducible
of dimension d, codimx(Y⊆X)=d−dimx(Y). When the ambient space X is fixed,
we will drop it from notation.
Definition 2.4**.**
For i≥1, the i-th successive minimum of L at x is defined by
[TABLE]
The definition makes sense, since codimxBs∣Ixt+L∣Q=0 for t≫0.
The infimum is a minimum, by Lemma 2.2. Note that
codimxBs∣Ixt+L∣Q is strictly less than i for t≥ϵi(L,x), and
at least i for 0≤t<ϵi(L,x). We obtain a chain of real numbers
[TABLE]
Note that codimxBs∣Ixt+L∣Q=0 for t≥ϵ1(L,x), and
codimxBs∣Ixt+L∣Q=i if and only if ϵi(L,x)>t≥ϵi+1(L,x).
Remark 2.5**.**
One may similarly define successive minima ϵi(R,x) in points x∈X,
associated to a graded subalgebra R⊆R(L). For example, when R is the image of the restriction
R(X′,L)→R(X,L′∣X), where X⊂X′ is a closed embedding and L′ is a Cartier divisor
on X′. The successive minima of such subalgebras appear in Lemma 4.1, for example.
Lemma 2.6**.**
Let L,L′ be Cartier divisors on X, let x∈X be a closed point and 1≤i≤d.
a)
ϵi(qL,x)=qϵi(L,x)* for every integer q≥1.*
b)
If L∼QL′, then ϵi(L,x)=ϵi(L′,x).
c)
Suppose ϵi(L,x)>0 and ϵi(L′,x)>0. Then
ϵi(L+L′,x)≥ϵi(L,x)+ϵi(L′,x).
Proof.
Property a) follows from Bs∣Ixt+(qL)∣Q=Bs∣Ixqt+L∣Q,
and b) from Bs∣Ixt+L∣Q=Bs∣Ixt+L′∣Q for all t≥0.
c) Let 0≤t<ϵi(L,x) and 0≤t′<ϵi(L′,x). Then both
Bs∣Ixt+L∣Q and Bs∣Ixt′+L′∣Q have codimension at x at least i. Since
[TABLE]
the codimension at x of the left hand side is at least i. Therefore t+t′<ϵi(L+L′,x).
Letting t and t′ approach ϵi(L,x) and ϵi(L′,x), respectively, we obtain the claim.
∎
In particular, we may define the i-th successive minimum of a Q-Cartier divisor L
at a point x∈X to be ϵi(L,x)=q1ϵi(qL,x), where q is a positive
integer such that qL is Cartier.
Lemma 2.7**.**
Let f:X′→X be a proper contraction of algebraic varieties, which is smooth
at a point x′∈X′. Let L be a Cartier divisor on X. Then
ϵi(L,f(x′))=ϵi(f∗L,x′) for all i.
Proof.
Denote L′=f∗L and x=f(x′). Since f is a contraction, f∗ induces isomorphisms
Γ(nL)→∼Γ(nL′)(n≥1).
Since f∗ maps Ixp into Ix′p, it induces injective homomorphisms
[TABLE]
These are also surjective if f is smooth at x′, since in this case the order of a section s
at x coincides with the order of f∗s at x′. We obtain isomorphisms
[TABLE]
Therefore Bs∣Ix′t+L′∣Q=f−1(Bs∣Ixt+L∣Q). Since f is smooth at x′,
we obtain
[TABLE]
Therefore ϵi(L,x)=ϵi(f∗L,x′) for all i.
∎
Lemma 2.8**.**
ϵi(L,x)>0* if and only if there exist integers p,q≥1 such that
codimxBs∣Ixp(qL)∣≥i. Moreover, in this case we have*
[TABLE]
Proof.
Denote ϵi=ϵi(L,x).
Step 1: codimxBs∣Ixp(qL)∣<i for every p,q≥1 such that p>qϵi.
Indeed, Bs∣Ixp(qL)∣ contains Bs∣Ixϵi+L∣Q, which has codimension at x strictly less than i.
Step 2: Let 0≤t<ϵi. Then there exist p,q≥1 with t<qp≤ϵi(L,x)
and codimxBs∣Ixp(qL)∣≥i.
Indeed, we have codimxBs∣Ixt+L∣Q≥i. There exists r≥1 such that
Bs∣Ixt+L∣Q=Bs∣Ix⌊rt⌋+1(rL)∣. Let q=r and p=⌊rt⌋+1.
By construction, t<qp. By the first step, qp≤ϵi.
If ϵi>0, there exist p,q≥1 such that codimxBs∣Ixp(qL)∣≥i, by Step 2.
Conversely, the latter implies ϵi≥qp>0, by Step 1. The two steps
give that ϵi is the supremum after all such qp.
∎
Lemma 2.9**.**
ϵ1(L,x)=0* if and only if one of the following holds:*
a)
κ(L)=−∞, or
b)
κ(L)=0* and every 0=s∈Γ(X,OX(nL))(n≥1) is invertible at x.*
Proof.
Suppose a) or b) holds, that is Γ(Ix(nL))=0 for n≥1. Therefore Bs∣Ix0+L∣Q=X.
We obtain ϵ1(L,x)=0.
Conversely, suppose ϵ1(L,x)=0. We may assume we are not in case a), that is
κ(L)≥0. Let n≥1 such that Γ(nL)=0. If dimkΓ(nL)≥2,
then Γ(Ix(nL))=0. This implies ϵ1(L,x)≥n1, a contradiction.
Therefore dimkΓ(nL)=1. Since again Γ(X,Ix(nL))=0, we have
Γ(nL)=ksn with sn(x)=0.
∎
In particular, ϵ1(L,x) is zero if κ(L)<0, and otherwise equals
[TABLE]
We call ϵ1(L,x) the width of L at the point x, also denoted by 0ptx(L).
If x∈X is a smooth point, then ϵ1(L,x) coincides with the width of L at the
geometric valuation induced by the exceptional divisor of the blow-up of X at x
(see [1, Section 2]).
By Lemma 2.8, ϵd(L,x)>0 if and only if there exist integers p,q≥1 such that
Bs∣Ixp(qL)∣={x} near x, and in this case, ϵd(L,x) is the supremum of
qp after all such p,q.
We call ϵd(L,x) the Seshadri constant of L at x, since we will see
later (Proposition 2.20)
that in the classical setting, it coincides with the Seshadri constant introduced by Demailly.
Lemma 2.10**.**
ϵd(L,x)>0* if and only if there exists n≥1 such that x∈/Bs∣nL∣ and the
induced rational map ϕ=ϕ∣nL∣:X⇢Xn⊆∣nL∣
satisfies ϕ−1ϕ(x)={x} near x. In particular, L is big.*
Proof.
Suppose ϵd(L,x)>0. Then there exists p,q≥1 such that
Bs∣Ixp(qL)∣={x} near x. By Lemma 1.4, there exists l≥1 such
that Bs∣qlL∣ is empty in a neighborhood of x. Denote n=ql. Let ϕ=ϕ∣nL∣:X⇢Xn⊆∣nL∣
be the induced rational map. It is regular on the open dense subset U=X∖Bs∣nL∣, which contains x.
Moreover, Bs∣Ix(nL)∣∩U=ϕ−1ϕ(x)∩U. Then
[TABLE]
Therefore ϕ−1ϕ(x)={x} near x, and dimXn=d. In particular, L is big.
Conversely, suppose ϕ=ϕ∣nL∣:X⇢Xn⊆∣nL∣ is regular near x
and ϕ−1ϕ(x)={x} near x. Then Bs∣Ix(nL)∣ equals {x} near x. Therefore ϵd(L,x)≥n1>0.
∎
The following lemma is a Successive minima version of [8, Proposition 6.4]
Lemma 2.11**.**
Let X be normal. Let x∈/Bs∣L∣Q. For n≥1 such that x∈/Bs∣nL∣, let
ϕ∣nL∣:X⇢Xn⊆∣nL∣ be the induced rational map. Let
μn:Yn→X be the normalization of the graph of ϕ∣nL∣.
In the mobile-fixed decomposition ∣μn∗(nL)∣=∣Mn∣+Fn, the mobile part is base point free.
Moreover, μn is an isomorphism over a neighborhood of x. Then
ϵi(L,x) is the supremum of nϵi(Mn,x), after all such n.
Proof.
Since x∈/Fn∈∣nμn∗L−Mn∣, we have Bs∣Ixt+(nμ∗L)∣Q⊆Bs∣Ixt+Mn∣Q
near x. Therefore ϵi(Mn,x)≤ϵi(nμ∗L,x)=nϵi(L,x).
Suppose now t<ϵi(L,x). There exist integers p,q≥1 with p>qt and
Bs∣Ixt+L∣Q=Bs∣Ixp(qL)∣. The identifications
Γ(Yq,OYq(Mq))=Γ(Yq,OYq(μq∗qL))=Γ(X,OX(qL))
induce an identification
Γ(Yq,Ix∈Yqp(Mq))=Γ(X,Ixp(qL)).
Therefore codimxBs∣Ixp(Mq)∣≥i.
We obtain p≤ϵi(Mq,x), that is t<qp≤qϵi(Mq,x).
The supremum is in fact a limit.
∎
Successive minima at a very general point
Contrary to the usual Seshadri constant, ϵi(L,x) is not lower semicontinuous with respect to x∈X in general.
For example, ϵ1(L,x) can be upper semicontinuous.
But one can show a weaker property, that is ϵi(L,x) is independent of the choice of a very general point x.
Definition-Proposition 2.12**.**
There exists a countable intersection of open dense subsets V⊆X such that
the correspondence V∋x↦ϵi(L,x) is constant. The common value does not depend
on V, and is denoted by ϵi(L).
Proof.
Let p1,p2:X×X→X be the natural projections, let δ:X→X×X be the diagonal
embedding, let Δ⊂X×X be the diagonal. Let p,q≥1 be integers.
Let Bpq be the locus where the composition
p2∗p2∗IΔp(qp1∗L)→IΔp(qp1∗L)⊂O(qp1∗L)
is not surjective.
By the definition, Bpq contains Δ. There exists an open dense subset Upq⊆X such that
the following properties hold:
p2∗(IΔp(qp1∗L))⊗k(x)→∼Γ(Ixp(qL)) for every x∈Upq.
2)
If Y is an irreducible component of Bpq which contains Δ, then p2:Y→X
is flat over Upq. In particular, Y∩p2−1(x) is equi-dimensional, of dimension dimY−d,
for every x∈Upq.
3)
δ(Upq) intersects only the irreducible components of Bpq which contain Δ.
Indeed, conditions 1) and 2) are open dense in X. For 3), we remove from X the δ-preimage of the
trace on Δ of the irreducible components of Bpq which do not contain Δ.
By 1), the natural inclusion Bs∣Ixp(qL)∣⊆Bpq∩p2−1(x) is an equality for every x∈Upq.
By 2) and 3), we have
[TABLE]
Let V=∩p,q≥1Upq. It is dense in X, a countable intersection of open subsets of X.
We claim that ϵi(L,⋅) is constant for x∈V. Suffices to show that if x∈V, t≥0 and ϵi(L,x)>t,
then ϵi(L,y)>t for every y∈V. Indeed, ϵi(L,x)>t implies that there exists qp>t
such that codimxBs∣Ixp(qL)∣≥i. That is codimδ(x)Bpq∩p2−1(x)≥i. From above, this
is equivalent to codim(Δ⊆Bpq)≥i. Arguing backwards, we see that codimyBs∣Iyp(qL)∣≥i
for every y∈V. Therefore ϵi(L,y)≥qp>t for every y∈V.
∎
Lemma 2.13**.**
Suppose ∣nL∣=∅ and dimϕ∣nL∣(X)≥i. Then ϵi(L)≥n1.
Proof.
Let Un⊆X be the open dense subset on which ϕ=ϕ∣nL∣:X⇢Xn⊆∣nL∣ is regular. We have
Bs∣Ix(nL)∣∩Un=ϕ−1ϕ(x)∩Un
for every x∈Un. We have dimxϕ−1ϕ(x)≥dimX−dimXn for all x∈Un, and
equality holds for an open dense subset Un′⊆Un. Let x∈Un′. Then
Suppose κ(L)≥i. There exists n≥1 such that ∣nL∣=∅, and
if X⇢Xn⊆∣nL∣ is the induced rational map,
then dimXn=κ(L)≥i. By Lemma 2.13, ϵi(L)≥n1.
Suppose κ(L)<i. Suppose Γ(nL)=0, let
ϕn:X⇢Xn⊆∣nL∣ be the induced rational map. For general x∈X we have
[TABLE]
Therefore ϵi(L,x)<n1. Letting n approach infinity, we obtain ϵi(L)=0.
∎
Let κ(L)=κ. If κ=0, then ϵi(L)=0 for all i. If κ≥1, we obtain
[TABLE]
Lemma 2.15**.**
Suppose κ(L′−L)≥0. Then ϵi(L)≤ϵi(L′) for all i.
Proof.
We may suppose ∣L′−L∣ is not empty. Choose C∈∣L′−L∣. Then Bs∣Ixp(qL′)∣⊆Bs∣Ixp(qL)∣∪SuppC.
Therefore Bs∣Ixt+L′∣Q⊆Bs∣Ixt+L∣Q∪SuppC. For x∈X∖C, we obtain
[TABLE]
By definition, ϵi(L,x)≤ϵi(L′,x) for all i.
∎
Lemma 2.16**.**
Let ϵi(L)>0, κ(L′)≥0, and ∣lL−l′L′∣=∅ for some integers l,l′≥1.
Then
[TABLE]
Proof.
Since κ(L′)≥0, we have ϵi(nL+L′)≥nϵi(L). Let C∈∣lL−l′L′∣. For n≥1, we have
[TABLE]
Therefore (l′n+l)ϵi(L)≥l′ϵi(nL+L′) by Lemma 2.15. We obtain
[TABLE]
Dividing by n and letting n approach +∞, we obtain the claim.
∎
Proposition 2.17**.**
Suppose X is projective. Let L≡L′ be numerically equivalent big Cartier divisors on X.
Then ϵi(L)=ϵi(L′) for every i.
Proof.
Let A be an ample divisor on X. Since (nL+A)−nL′ is ample, ϵi(nL+A)≥nϵi(L′).
By Lemma 2.16, we obtain ϵi(L)≥ϵi(L′). The other inequality holds by the same argument.
∎
In particular, if X is projective and L is big, ϵ1(L)
coincides with the invariant m(L) introduced by Nakamaye [17, page 2].
We may rephrase [7, Proposition 2.3] and [18, Lemma 1.3] as follows:
Lemma 2.18**.**
Let X be a proper algebraic variety, let L be a Cartier divisor, let t≥0.
Let x∈X be a very general point and Z⊆Bs∣Ixt+L∣Q an irreducible
component containing x.
Then for integers p,q≥1 and s∈Γ(Ixp(qL)), we have ordZ(s)≥p−qt.
Proof.
Let p1,p2:X×X→X be the natural projections, let δ:X→X×X be the
diagonal embedding, let Δ⊂X×X be the diagonal.
Let p,q≥1 be integers with p>qt.
Let Bpq be the locus where the composition
p2∗p2∗IΔp(qp1∗L)→IΔp(qp1∗L)⊂O(qp1∗L)
is not surjective. There exists an open dense subset Xpq⊆X such that
[TABLE]
We deduce that Bs∣Ixp(qL)∣⊆Bpq∩p2−1(x) for all x∈X, and equality holds for x∈Xpq.
Define B=∩p>qtBpq, a closed subset of X×X. Denote X0=∩p>qtXpq,
a countable intersection of open dense subsets of X. Then
Bs∣Ixt+L∣Q⊆B∩p2−1(x) for all x∈X, and equality holds for x∈X0.
We have Δ⊆B, so at least one irreducible component of B contains Δ.
Let B′ be the union of irreducible components of B which do not contain Δ.
Then δ−1(Δ∩B′) is a proper closed subset of X.
Its complement U⊆X is an open dense subset. For every x∈U, an irreducible component of B which
contains δ(x) must also contain Δ. We may further shrink U, so that U is smooth.
Let x∈U∩X0. Let Z be an irreducible component of Bs∣Ixt+L∣Q which contains x,
let 0=s∈Γ(Ixp(qL)).
There exists an irreducible component Z~ of B, which contains δ(x), such that Z⊆Z~∩p2−1(x).
Since Z~ contains δ(x) and x∈U, Z~ contains Δ.
Since x∈Xpq, there exists s~∈Γ(X×T,IΔp(qp1∗L)) with
s~∣p2−1(x)=s, where T is a neighborhood of x in U, which we may suppose affine and smooth.
Let m=ordZ~s~. Since Z~ contains Δ, p1:Z~→X is dominant.
Let Dm be a general differential operator of order at most m on T. Then
Dms~∈Γ(X×T,IΔp−m(qp1∗L)) does not vanish along Z~∩X×T,
by the argument of [7, Proposition 2.3] (the argument goes through even if X is singular, since
T is affine smooth and X×T is smooth at the general point of Δ∩X×T).
Therefore Z~∩X×T is not contained Bp−m,q. Therefore p−m≤qt.
We conclude ordZ(s)=ordZ(s~∣p2−1(x))≥ordZ(s~)≥ordZ~(s~)=m≥p−qt.
∎
Seshadri constant as d-th minimum
Recall our notation: X is a proper algebraic variety of dimension d, L is a Cartier divisor on X, and
x∈X is a closed point.
Proposition 2.19** (Seshadri criterion).**
The divisor L is ample if and only if ϵd(L,x)>0 for any x∈X.
Proof.
Suppose L is ample. There exists n≥1 such that nL is very ample. That is,
Ix(nL) is generated by global sections, for every point x∈X. Then
Bs∣Ix(nL)∣={x}, hence ϵd(L,x)≥n1 for every x.
Conversely, suppose ϵd(L,x)>0 for any x∈X. To show that L is ample,
by Nakai-Moishezon’s criterion, suffices to show that for every irreducible subvariety Y⊆X,
of dimension r≥1, we have (L∣Y)r>0. Note that 0<ϵd(L,x)≤ϵr(L∣Y,x) for every
x∈Y. If r<d=dimX, we conclude by induction that L∣Y is ample, hence (L∣Y)r>0.
It remains to deal with the case Y=X. Since ϵd(L,x) is positive at a very general point,
L is big by Lemma 2.14. Since L is also nef by induction, we deduce (Ld)>0.
∎
Proposition 2.20**.**
Suppose X is projective and L is nef.
Then
[TABLE]
Denote by ϵs(L,x) the infimum after all integral curves passing through x, and by
ϵj(L,x) the supremum in the claim.
First, we show the following lemma:
Lemma 2.21**.**
For a Cartier divisor L on a proper X,
a)
ϵd(L,x)≥ϵj(L,x),
b)
*if L is nef, ϵs(L,x)≥ϵd(L,x)≥ϵj(L,x).
*
Proof.
a) Let p,q≥1 such that
Γ(qL)→Ox/Ixp+1 is surjective. Equivalently, Ixi(qL) is globally generated
at x for every 0≤i≤p. Then Bs∣Ixp(qL)∣={x} near x. Therefore qp≤ϵd(L,x).
Taking the supremum after all qp, we obtain ϵd(L,x)≥ϵj(L,x).
b) Suppose L is nef.
If ϵd(L,x)=0, ϵs(L,x)≥ϵd(L,x) is clear.
Assume ϵd(L,x)>0 and let 0<t<ϵd(L,x). Then Bs∣Ixt+L∣Q={x}
near x. Let C⊆X be an integral curve passing through x. Since C is not contained
in Bs∣Ixt+L∣Q, there exists n≥1 and D∈∣nL∣ such that ordx(D)>nt and C is not
contained in the support of D. Then
[TABLE]
Therefore t≤ϵs(L,x). Taking the supremum after all such t, we obtain ϵs(L,x)≥ϵd(L,x).
∎
To prove Proposition 2.20, it is enough to show ϵs(L,x)≤ϵj(L,x) by Lemma 2.21.
In fact, ϵs(L,x)=ϵj(L,x) is well known at least for ample L and smooth x∈X (see [13, Theorem 5.1.17]).
Let f:Y→X be the blow-up of X at x, with
exceptional divisor E. Note that f∗L−ϵs(L,x)E is nef, and ϵs(L,x)≥0 is maximal
with this property.
Let A be an ample divisor on X.
The augmented base locus B+(L) of L is defined by
[TABLE]
which does not depend on the choice of A.
By [15], [4],
[TABLE]
holds,
where the union runs over the subvarieties V⊆X such that L∣V is not big.
First,
assume x∈B+(L).
Then there exists a subvariety V⊆X containing x such that L∣V is not big.
Since L is nef, we have (LdimV.V)=0.
Hence ϵs(L,x)=0 follows from
[TABLE]
where V′⊆Y is the strict transform of V.
By Lemma 2.21 b), all three invariants equal to [math].
So we may assume x∈B+(L).
If we replace A with its suitable multiple,
we can take l≥1 and an effective Cartier divisor D such that lL∼A+D and x∈SuppD.
Since A is ample and −E is f-ample, there exists an integer a≥1
such that af∗A−E is ample.
By Lemma 2.21 b), it suffices to show ϵs(L,x)≤ϵj(L,x), which follows from the following claim:
if p,q≥1 are integers such that p−1≤(q−la)ϵs(L,x), then
Γ(nqL)→Ox/Ixnp+1 is surjective for n≫0.
Indeed, the divisor
[TABLE]
is ample. By Serre vanishing, H1(Y,IE(f∗(nqL−naD)−npE))=0 for n≫0.
We obtain an exact sequence
[TABLE]
Since n≫0, the hypothesis of Lemma 1.6 are satisfied. Therefore
Γ(nqL−naD)→Ox/Ixnp+1 is surjective for n≫0.
Since D is effective and away from x, we deduce that Γ(nqL)→Ox/Ixnp+1 is surjective.
Therefore qp≤ϵj(L,x). Since qp can be arbitrary close to ϵs(L,x), we obtain ϵs(L,x)≤ϵj(L,x).
∎
The equalities in Proposition 2.20 hold on proper X if L is semiample:
Lemma 2.22**.**
Suppose X is proper and L is semiample. Then
[TABLE]
Proof.
There exists a proper contraction f:X→Y, with Y projective, such that
nL∼f∗A for some integer n≥1 and some ample divisor A on Y. Let Exc(f)
be the exceptional locus of f, that is the locus of points x∈X such that dimxf−1f(x)>0.
If x∈Exc(f), we see that ϵs(L,x)=ϵd(L,x)=ϵj(L,x)=0. Suppose
x∈/Exc(f). Then dimxf−1f(x)=0. By Lemma 1.2, f is an isomorphism over a
neighborhood of f(x).
It follows that ϵd(nL,x)=ϵd(A,f(x)) and ϵ∗(nL,x)=ϵ∗(A,f(x)) for ∗=s,j.
We are reduced to the ample case, so we conclude from Proposition 2.20.
∎
Remark 2.23**.**
By Lemma 2.22, the Seshadri constant of a semiample divisor L at a point x∈X defined by
Demailly [5, Theorem 6.4] coincides with ϵd(L,x).
Combining with Lemma 2.11, we deduce that if X is normal, projective, L is big,
and x is a point away from the augmented base locus of L,
then ϵd(L,x) coincides with the moving Seshadri constant ϵm(x,L) introduced by Nakamaye [16, Definition 0.4] (see also [8, Section 6]).
The interpretation of the Seshadri
constant in terms of jets generation can be sharpened as follows: if ϵd(L,x)=ϵ>0, there
exist constants c1(x)≥0 and c2(x)≥0 such that Γ(qL)→Ox/Ixp+1 is surjective for all integers
p,q≥1 such that p≥c1(x) and qϵ−p≥c2(x). This statement reduces again to the
ample case, which is the following Lemma.
Lemma 2.24**.**
Suppose L is ample. Let
ϵ=infC∋xmultx(C)(L⋅C)
the Seshadri constant of L at x. Then there exist constants c1,c2≥0 such that for every
integers p,q≥1 with p≥c1 and qϵ−p≥c2, the jet map Γ(qL)→OX/Ixp+1
is surjective.
Proof.
Let f:Y→X be the blow-up of X at x, with exceptional divisor E.
Since L is ample and −E is f-ample,
there exists an integer a≥1 such that af∗L−E is ample. By Nakai’s ampleness criterion, qf∗L−pE is
ample if and only if qϵ>p>0. We have
[TABLE]
For (q−ma)ϵ≥p−m≥0, the divisor (q−ma)f∗L−(p−m)E is nef. By Fujita’s extension of Serre vanishing
(see [13, Theorem 1.4.35]), there exists an integer m=m(OY,af∗L−E) such that H1(qf∗L−pE)=0
for (q−ma)ϵ≥p−m≥0.
We have Ixp+1=f∗(IEp+1) and R1f∗(IEp+1)=0 for p≥c′(x). Set c1=max(c′(x),m−1)
and c2=maϵ−m+1.
Let p,q≥1 be integers with p≥c1 and qϵ−p≥c2. Then p≥c′(x) and (q−ma)ϵ≥p+1−m≥0.
The latter inequality implies that we have a surjection
[TABLE]
From the former inequality and Lemma 1.6, we deduce that Γ(qL)→OX/Ixp+1 is surjective.
∎
Successive minima in terms of jets generation
Once again, X is a proper algebraic variety, L is a Cartier divisor on X, and x∈X is a closed point.
The following proposition is proved in [8, Proposition 6.6] for the moving Seshadri constant on smooth X:
Proposition 2.25**.**
We have ϵd(L,x)>0 if and only if there exist p,q≥1 such that Γ(qL)→Ox/Ixp+1
is surjective. Moreover, in this case we have
[TABLE]
Proof.
Denote by ϵj(L,x) the supremum in the claim.
We obtain ϵd(L,x)≥ϵj(L,x) by Lemma 2.21 a).
For the converse,
we show the following claim:
Claim 2.26*.*
If Bs∣Ixr(L)∣={x} near x for some integer r>0, then r≤ϵj(L,x).
Proof.
We use the notation in Lemma 1.5.
We identify x with the point g(f−1(x))∈Y.
Recall that
we have a natural homomorphism
Γ(IZq(qL))→Γ(qA),
which is an isomorphism for q≥c(IZ).
Since Bs∣Ixr(L)∣={x} near x,
Bs∣Ixr(A)∣={x} near x∈Y and hence r≤ϵd(A,x) holds.
By applying Proposition 2.20 to the ample divisor A,
ϵd(A,x)=ϵj(A,x) holds.
Hence, for any 0<t<r,
we can take t<qp<r such that Γ(qA)→Ox,Y/Ixp+1 is surjective.
We note that we can take arbitrary large such p,q by Lemma 2.24.
So we take such p,q with q≥c(IZ).
Since q≥c(IZ), we have an isomorphism Γ(IZq(qL))→∼Γ(qA).
Hence Γ(IZq(qL))→Ox,X/Ixp+1 is surjective since
so is Γ(qA)→Ox,Y/Ixp+1.
Then Γ(qL)→Ox,X/Ixp+1 is also surjective since x∈Z,
and hence t<qp≤ϵj(L,x).
Letting t approach r,
we obtain the claim.
∎
For any 0≤t′<ϵd(L,x),
we can take t′<sr<ϵd(L,x) such that Bs∣Ixr(sL)∣={x} near x.
Applying the above claim to sL,
it holds that
[TABLE]
where the last inequality follows from the definition of ϵj.
Hence t′<sr≤ϵj(L,x) and the proposition follows by letting t′ approach ϵd(L,x).
∎
Lemma 2.27**.**
Suppose ϵ1(L,x)>0. Then
[TABLE]
Proof.
Note that Γ(Ixp(qL))→Ixp/Ixp+1 is non-zero if and only if there exists s∈Γ(qL) such that
ordx(s)=p. In particular, ϵ1(L,x)≥qp. On the other hand, let p,q≥1 such that Γ(Ixp(qL))=0.
Let p′≥p be maximal such that Γ(Ixp′(qL))=0. There exists s∈Γ(qL) such that ordx(s)=p′.
Therefore qp≤qp′≤ϵ1(L,x). We conclude by Lemma 2.8.
∎
Lemma 2.28**.**
Suppose ϵd(L,x)>0. Then
[TABLE]
Proof.
The inequality ≤ follows from Proposition 2.25. For the opposite inequality, suppose that
Γ(Ixp(qL))→Ixp/Ixp+1 is surjective. By Nakayama’s Lemma, Ixp(qL)
is generated by global sections at x. Therefore Bs∣Ixp(qL)∣={x} near x.
Therefore p/q≤ϵd(L,x).
∎
3. Volume versus the successive minima
The following well known statement may be considered as the analogue of
Minkowski’s first main theorem:
Proposition 3.1**.**
Let X be proper of dimension d, let L be a Cartier divisor on X, let x∈X be a closed point. Then
vol(L)≤multx(X)⋅ϵ1(L,x)d.
Proof.
Let t>ϵ1(L,x) be a real number. Let n≥1.
We have Γ(Ix⌊nt⌋+1(nL))=0, that is the jet map
[TABLE]
is injective. Therefore h0(nL)≤dimk(Ox/Ix⌊nt⌋+1). For n≫0,
the right hand side equals P(⌊nt⌋), where P(T) is a polynomial with leading term
d!eTd, where e=multx(X). Therefore
[TABLE]
Letting t approach ϵ1(L,x), we obtain the claim.
∎
Lemma 3.2**.**
Let X be a proper variety of dimension d, let ϵi=ϵi(L,x) be the
Seshadri successive minima of a Cartier divisor L at a closed point x. Then
[TABLE]
In particular, multx(X)⋅ϵd(L,x)d≤vol(L).
Proof.
Suffices to show that for any real numbers 0≤ti<ϵi, we have
[TABLE]
To prove this inequality, note first that there exist D1,…,Dd∈∣L∣Q such that
ordx(Di)>ti and ∩i=1dDi={x} near x. Indeed, there exists
D1∈∣L∣Q with ordx(D1)>t1. Since codimxBs∣Ixt2+L∣Q≥2,
no irreducible component of D1 through x is contained in Bs∣Ixt2+L∣Q.
There exists D2∈∣L∣Q with ordx(D2)>t2 and codimx(D1∩D2)=2.
Since codimxBs∣Ixt3+L∣Q≥3, no irreducible component of D1∩D2 through
x is contained in Bs∣Ixt3+L∣Q.
Therefore there exists D3∈∣L∣Q with ordx(D3)>t3 and codimx(D1∩D2∩D3)=3.
Iterating the construction, we obtain the chain of divisors in d steps.
There exists an integer q≥1 such that qDi are the zero divisors of some non-zero global sections
si∈Γ(X,Ix⌊qti⌋+1(qL)).
Let Z be the (possibly empty) subscheme ∩i=1dqDi∖{x}⊂X.
Let f:Y→X be the
blow-up of X along IZ, let E be the exceptional divisor on Y. Then f∗s1,…,f∗sd become
global sections of OY(qf∗L−E), having no common zeroes near E. Therefore
Bs∣qf∗L−E∣⊆{x} (we identify x with a point of Y). By Lemma 1.4,
M=qf∗L−E is semiample.
Set Di′=f∗qDi−E∈∣M∣.
By construction,
∩i=1dDi′={x}⊂Y as sets.
Hence we have
[TABLE]
by [10, page 233],
where i(x,D1′⋅D2′⋯Dd′;Y) is the intersection multiplicity.
For l≥c(IZ),
the natural homomorphism IZl→f∗OY(−lE) is an isomorphism,
hence so is
[TABLE]
Therefore qdvol(L)=vol(qL)≥vol(M)=(Md)>multx(X)⋅∏i=1dqti. Dividing out by q and letting ti approach ϵi, we obtain the claim.
∎
Corollary 3.3**.**
Let X be an algebraic variety of dimension d, let x be a closed point and L a Cartier divisor on X.
Then
[TABLE]
Minkowski’s second main theorem for Seshadri successive minima
Lemma 3.4**.**
Let L be a Cartier divisor on a proper algebraic variety X.
Let ϵi=ϵi(L) be the Seshadri successive minima of L at a very general
point. Then
[TABLE]
Proof.
Let x∈X be a very general point.
Step 1: Let p≥0 be an integer. Then
[TABLE]
Indeed, we have an exact sequence
[TABLE]
Therefore h0(Ixp(L))−h0(Ixp+1(L))=dimkIm(r). We may suppose Im(r)=0.
In particular, h0(Ixp(L))>0, hence p≤ϵ1.
If p≤ϵd, the desired inequality becomes dimIm(r)≤∣Nd(p)∣, which follows from the inclusion
Im(r)⊆Γ(OPd−1(p)). Therefore we may suppose ϵd<p≤ϵ1.
Denote by J the set of indices 1<i≤d such that ϵi<p and ϵi<ϵi−1.
Fix i∈J. Since ϵi<ϵi−1, Bs∣Ixϵi+L∣Q has codimension i−1 at x.
Choose an irreducible component Zi−1, of codimension i−1, passing through x. On the blow-up of X at x,
the proper transform of Zi−1 has in common with the exceptional locus E≃Pd−1 at least one irreducible
subvariety Wi−1⊂Pd−1, of codimension i−1.
Let s∈Γ(Ixp(L)). By Lemma 2.18, ordZi−1(s)≥p−ϵi. Therefore the image of r
is contained in {P∈Γ(OPd−1(p));ordWi−1(P)≥p−ϵi}. We conclude
[TABLE]
By Proposition 1.7, the right hand side has dimension at most
Indeed, h0(L)−h0(Ixp+1(L))=∑l=0ph0(Ixl(L))−h0(Ixl+1(L)).
Applying Step 1 to each term of the sum, we obtain the desired inequality.
Step 3) Set p=⌊ϵ1⌋ in Step 2.
Then p+1>ϵ1, so h0(Ixp+1(L))=0. We obtain
[TABLE]
Since Zd∩□(⌊t1⌋,…,⌊td⌋)=Zd∩□(t1,…,td),
we obtain
[TABLE]
∎
Proposition 3.5**.**
Let L be a Cartier divisor on a proper algebraic variety X, with Iitaka dimension κ(L)=κ≥1.
Let ϵi=ϵi(L) be the Seshadri successive minima of L at a very general
point. Then
[TABLE]
In particular, if L is big, vol(L)≤d!⋅vol□(ϵ1,…,ϵd).
Proof.
Suppose tκ+1=⋯=td=0. Then □(t1,…,td)=□(t1,…,tκ)×0,
where [math] is the origin in Rd−κ. We have ϵ1≥⋯≥ϵκ>0=ϵκ+1=⋯=ϵd. Since ϵi(nL)=nϵi(L), we obtain
[TABLE]
As n→∞, the right hand side grows like vol□(ϵ1,…,ϵκ)⋅nκ+O(nκ−1).
Therefore the claim holds.
∎
Proposition 3.5 generalizes the following result of
Nakamaye [17, Proof of Corollary 3]: if L is an ample
divisor on a smooth projective surface, then (L2)≤2ϵ1(L)ϵ2(L)−ϵ2(L)2.
Combining Lemma 1.8, Propositions 3.2 and 3.5, we obtain the following equivalent of Minkowski’s
second theorem for successive minima:
Theorem 3.6**.**
Let L be a big Cartier divisor on a d-dimensional proper algebraic variety X. Then
[TABLE]
So the volume of L is essentially the product of the successive Seshadri minima of L at a
very general point.
Remark 3.7**.**
If (X,L)=(Pd,O(1)), ϵi(L)=1 for any i. Hence ∏i=1dϵi(L)vol(L)=1.
On the other hand,
consider (X,L)=((P1)d,O(w1,…,wd)), where w1≥w2≥⋯≥wd are positive integers.
Then ϵi(L)=∑j=idwj
and
vol(L)=d!∏i=1dwi
as we will see in Example 4.5.
Hence
[TABLE]
could be arbitrary close to d!
if we take w1≫w2≫⋯≫wd.
Thus the lower and upper bounds in Theorem 3.6 are sharp.
We also note that the upper bound is not attained for d≥2
since vol□(ϵ1(L),…,ϵd(L))<∏i=1dϵi(L) by □(ϵ1(L),…,ϵd(L))⊊[0,ϵ1(L)]×□(ϵ2(L),…,ϵd(L)).
4. Succesive minima on toric varieties
For standard terminology on toric varieties, the reader may consult [19].
Let X=TNemb(Δ) be a proper toric variety (normal), of dimension d.
Let L be a Cartier divisor on X. Modulo linear equivalence, we may suppose L
is torus invariant. Due to the torus action, ϵi(L,⋅) is constant on TN⊂X.
Therefore ϵi(L)=ϵi(L,1), where 1 denotes the unit of the torus TN.
We will estimate ϵi(L,x) when either x is a closed invariant point, or x=1.
At a closed invariant point
Let x∈X be a closed invariant point. It corresponds to a top cone σ∈Δ(top),
and we obtain an open affine neighborhood x∈Uσ=Speck[M∩σ∨].
Denote by S the semigroup M∩σ∨∖0, so that
I(x∈Uσ)=⊕m∈Sk⋅χm.
For p≥1, denote S(p)={s1+⋯+sp;s1,…,sp∈S}. Therefore
I(x∈Uσ)p=⊕m∈S(p)k⋅χm.
Since L is Cartier, there exists u∈M such that (χu)+L∣Uσ=0. We have
□L−u⊆σ∨.
The subspace {s∈Γ(X,OX(qL));ordx(s)≥p}⊆Γ(X,OX(qL))
is torus invariant. Therefore a basis over k consists of monomials χm such that
m∈M∩q□L and m−qu∈S(p).
Let P=Conv(S+σ∨) be the Newton polytope associated to S⊂σ∨.
Let B be the closure of the complement σ∨∖P. Then B is compact and
contains a relatively open neighborhood of the origin in σ∨. In particular,
σ∨=∪t≥0tB. Note that B may not be convex, but in case x is a smooth point,
B is a unit simplex. Moreover, d!⋅volM(B)=multx(X).
We claim that S(p)⊆M∩pP⊆S(p−d+1). Indeed, the first inclusion is clear.
For the second, let m∈M∩pP. Then m=∑i∈Itimi+m′, where ti≥0, mi∈S,
∑iti=p and m′∈σ∨, and the cardinality of I is at most d
(use Carathéodory’s theorem). Then m=∑i⌊ti⌋mi+(m′+∑i{ti}mi),
and ∑i⌊ti⌋>∑i(ti−1)=p−d. Therefore m∈S(p−d+1).
We obtain
Γ(Ixp(qL))⊆⊕m∈M∩q□L∩pP+quk⋅χm⊆Γ(Ixp−d+1(qL)).
Therefore the two algebras
[TABLE]
have the same stable base locus. Denote t+P=∩ϵ>0(t+ϵ)P. We deduce that
near x,
Bs∣Ixt+L∣Q is the intersection of Supp(χm), where
m∈(□L−u)∩t+P∩MQ and (χm) is the effective Q-divisor defined by χm.
Lemma 4.1**.**
Near x, Bs∣Ixt+L∣Q is the union of invariant closed irreducible subvarieties x∈Y⊆X
such that 0ptx(R(L)∣Y)≤t. If L is ample, the latter inequality means 0ptx(L∣Y)≤t.
Proof.
Bs∣Ixt+L∣Q is the intersection of all Supp(χm),
where χm∈Γ(OX(qL)) with ordx(χm)>qt.
In particular, Bs∣Ixt+L∣Q is torus invariant.
Let x∈Y⊆X be a torus invariant subvariety.
By definition,
0ptx(R(L)∣Y)>t if and only if the subspace
[TABLE]
contains a non-zero element for some q.
This subspace is torus invariant, and hence has a basis consists of monomials.
Thus this subspace has a non-zero element if and only if there exists χm∈Γ(OX(qL)) with χm∣Y=0 and ordx(χm∣Y)>qt.
For monomials, the restriction to Y does not change the order,
that is, ordx(χm∣Y)=ordx(χm) if χm∣Y=0.
Hence 0ptx(R(L)∣Y)>t if and only if there exists χm∈Γ(OX(qL)) with χm∣Y=0 and ordx(χm)>qt for some q,
which is equivalent to say that Y is not contained in Bs∣Ixt+L∣Q.
If L is ample, the restriction map Γ(qL)→Γ(qL∣Y) is surjective for any q, and therefore 0ptx(R(L)∣Y)=0ptx(L∣Y).
∎
Lemma 4.2**.**
Let x∈Y⊆X be an invariant closed subvariety corresponding to a face τ≺σ.
Then 0ptx(R(L)∣Y)=min{t≥0;(□L−u)∩τ⊥⊆tB∩τ⊥}∈Q.
In particular,
0ptx(L)=min{t≥0;□L−u⊆tB}∈Q.
Proof.
By Lemma 4.1,
0ptx(R(L)∣Y)≤t if and only if Y⊆Bs∣Ixt+L∣Q, if and only if (□L−u)∩t+P∩τ⊥∩MQ=∅.
Since □L,u,P are rational, this condition is equivalent to (□L−u)∩t+P∩τ⊥=∅, which is
equivalent to □L−u∩τ⊥⊆tB∩τ⊥.
∎
We obtain the following proposition which generalizes [3, Corollary 4.2.2].
Proposition 4.3**.**
ϵi(L,x)* is the minimum of 0ptx(R(L)∣Y),
after all closed irreducible invariant subvarieties x∈Y⊆X of codimension i−1.
In particular, ϵi(L,x)∈Q.*
Suppose L is ample. Then 0ptx(R(L)∣Y)=0ptx(L∣Y), and therefore
ϵi(L,x) is the minimum of 0ptx(L∣Y), after all closed irreducible invariant subvarieties
x∈Y⊆X of codimension i−1. In particular,
ϵd(L,x)=min{(L⋅C);x∈C⊆X invariant curve}.
Moreover, we have inclusions
[TABLE]
and ϵd,ϵ1 are maximal and minimal, respectively, with this property.
At a general point
Let □⊂MR be the moment polytope of L. Then □−□⊂MR
is a [math]-symmetric compact convex set, of dimension κ=κ(L). The Minkowski
successive minima of (M,□−□) are
[TABLE]
We have 0≤λ1≤λ2≤⋯≤λκ<λκ+1=+∞.
The polar convex set (□−□)∗⊂NR coincides with the set of linear functionals
φ∈NR such that lengthφ(□)≤1. It is unbounded if and only if κ<rankM.
The Minkowski successive minima of (N,(□−□)∗) are
[TABLE]
Example 4.4**.**
Consider (X,L)=(Pd,O(w)), where w is a positive integer. Since the ambient
is homogeneous, ϵi(L,x)=ϵi(L) for all x∈X and i. Then
[TABLE]
Let e1,…,ed be the standard basis of Zd. The moment polytope of L is
[TABLE]
The difference □−□ is the convex hull of ±wei(1≤i≤d),±w(ei−ej)(1≤i<j≤d),
which is contained in {∑i=1dxiei;∣xi∣≤w,∣xi+xj∣≤w}. We compute
[TABLE]
Let e1∗,…,ed∗ be the dual basis of Zˇd. The polar body (□−□)∗ is
{∑i=1dxi∗ei∗;∣xi∗∣≤1/w,∣xi∗−xj∗∣≤1/w}. We compute
[TABLE]
Example 4.5**.**
Consider (X,L)=((P1)d,O(w1,…,wd)), where w1≥w2≥⋯≥wd are positive integers.
Since the ambient is homogeneous, ϵi(L,x)=ϵi(L) for all x and i.
Let x=[1:0]d∈X. Then Bs∣Ixt+L∣Q consists of the invariant cycles
Y through x (affine spaces with coordinates zi(i∈I)) such that ∑i∈Iwi≤t. Therefore
ϵi(L)=min∣I∣=d−i+1∑j∈Iwj.
Since we ordered the weights, we obtain
[TABLE]
The volume is
[TABLE]
The moment polytope of L is □=∏i=1d[0,wi]⊂Rd. Then
□−□=∏i=1d[−wi,wi], so
[TABLE]
The polar body (□−□)∗ is {x∈Rˇd;∑i=1d∣xi∣wi≤1}. Therefore
[TABLE]
We obtain
[TABLE]
Lemma 4.6**.**
ϵj(L)⋅λj≥1.
Proof.
Denote μ=1/λj(M,□−□). There exist u1,…,uj∈M, primitive and
linearly independent, such that μui=mi′−mi for some mi,mi′∈□.
The inclusion [mi,mi′]⊂□ induces a dominant rational map
[TABLE]
Therefore there exists μFi+Di∈∣L∣Q, where Fi is the fiber of φi through 1,
and Di is an effective invariant divisor on X. We have Fi∩TN=TN∩ui⊥.
Since ui are linearly independent, we have codim1∩i=1jFi=j.
We obtain codim1Bs∣I1μL∣Q≥j. Therefore μ≤ϵj(L).
∎
Lemma 4.7**.**
ϵd−j+1(L)≤j⋅λj∗.
Proof.
Suppose λj∗=λ>0. There exist
φ1,…,φj∈N∩λ⋅(□−□)∗, linearly independent.
The property φi∈λ⋅(□−□)∗ means that the interval φi(□)
has length at most λ. Consider the induced homomorphism of lattices
[TABLE]
whose image has rank j since φi are linearly independent.
Then □ is mapped onto a polytope of dimension j, contained in
∏i=1j[xi,xi+λ] for some x1,…,xj∈R.
The lattice homomorphism induces a dominant rational map X⇢Yd−j whose
fiber (F,L∣F) through x=1 has dimension j, and is dominated by (P1,OP1(λ))j.
Therefore
[TABLE]
by Example 4.5.
Let D∈∣I1jλ+L∣Q. Then D∣F∈∣I1jλ+(L∣F)∣Q, which is empty since
ϵ1(L∣F,1)≤jλ. Therefore D∣F=0. We deduce
[TABLE]
Therefore codim1Bs∣I1jλ+L∣Q≤d−j<d−j+1. Then ϵd−j+1(L)≤jλ.
∎
Remark 4.8**.**
For ϵd(L),
Lemmas 4.6, 4.7 state that λd−1≤ϵd(L)≤λ1∗.
These inequalities also follow from [11, Theorem 3.6].
Theorem 4.9**.**
The invariants ϵi(L),1/λi,λd−i+1∗ are all equivalent. More precisely,
[TABLE]
Proof.
Use Banaszczyk’s bound [2] in Mahler’s transference theorem 1≤λiλd−i+1∗≤d for the
second inequality, and the two lemmas above.
∎
Theorem 4.10**.**
The Seshadri constant of L at a very general point
is proportional to the lattice width of the moment polytope □L. More precisely,
[TABLE]
Proof.
If L is not big, both invariants are zero. Suppose L is big.
Then ϵ(L)=ϵd(L) and 0pt(□L)=λ1∗, and we can apply the above theorem.
∎
Remark 4.11**.**
For toric varieties, we may establish the inequalities
[TABLE]
of Theorem 3.6 as follows: the left hand side inequality is easy to see.
For the right hand side, recall the second main theorem of Minkowski,
in the stronger form due to Davenport-Estermann:
[TABLE]
We have vol(L)=d!volM(□). From ϵiλi≥1, we obtain
∏iϵi⋅∏iλi≥1. Therefore vol(L)≤d!∏iϵi.
5. Adjoint linear systems and the Flatness Theorem of Khinchin
Recall first results of Demailly [5] and Ein, Küchle, Lazarsfeld [7]:
Theorem 5.1**.**
Let X be a smooth projective variety of dimension d, let L be a nef and big
Q-Cartier divisor on X whose fractional part has normal crossing support. Let ϵ be the Seshadri constant of L at a very general
point x∈X, which coincides with ϵd(L,x). The following properties hold:
The jet map Γ(⌈KX+L⌉)→Ox/Ix1+⌈ϵ−d−1⌉
is surjective. In particular,
[TABLE]
2)
If ϵ>d, then Γ(⌈KX+L⌉)=0.
3)
If ϵ>d+1, then ∣⌈KX+L⌉∣ maps X onto a variety of dimension d.
4)
If ϵ>2d, then ∣⌈KX+L⌉∣ maps X birationally onto a variety of dimension d.
Proof.
Let f:Y→X be the blow-up at x, with exceptional divisor E.
Since L has integer coefficients near x, ⌈KY+f∗L⌉=f∗⌈KX+L⌉+(d−1)E.
We may suppose p=⌈ϵ−d−1⌉ is non-negative. Then
⌈KY+f∗L−(p+d)E⌉=f∗⌈KX+L⌉−(p+1)E
and
[TABLE]
By the Leray spectral sequence, the natural homomorphism
[TABLE]
is injective.
But f∗L−(p+d)E=f∗L−(⌈ϵ⌉−1)E is nef and big, since ⌈ϵ⌉−1<ϵ.
By Kawamata-Viehweg vanishing, H1(Y,⌈KY+f∗L−(p+d)E⌉)=0. Therefore H1(X,Ixp+1(⌈KX+L⌉))=0.
Then Γ(⌈KX+L⌉)→Ox/Ixp+1 is surjective. Since x is a smooth point, the
right hand side has dimension (dp+d).
This follows from 1).
The 1-jet map Γ(⌈KX+L⌉)→Ox/Ix2 is surjective.
Therefore ∣⌈KX+L⌉∣ moves, and the induced rational map is generically finite.
Let x,y be two very general points, let f:Y→X be the blow-up at x,y. Then
[TABLE]
is nef and big. By Kawamata-Viehweg vanishing, H1(Y,⌈KY+f∗L−dEx−dEy⌉)=0.
But ⌈KY+f∗L−dEx−dEy⌉=f∗⌈KX+L⌉−Ex−Ey. Therefore H1(X,Ix⊗Iy(⌈KX+L⌉))=0.
∎
We generalize the flatness theorem of Khinchin. The original statement
says that a convex body which contains no lattice points must have lattice width bounded
above by a constant which depends only on the dimension (see [12] and part a) of the theorem below).
We show that the same conclusion holds if the lattice points of the convex body are degenerate (part b)
of the theorem below).
Theorem 5.2**.**
Let M≃Zd be a lattice, let □⊂MR be a compact convex set, of dimension d.
Let w be the lattice width of □ with respect to M.
a)
If w>d2, then M∩int□=∅.
b)
If w>d(d+1), then dim(M∩int□)=d.
c)
If w>2d2, then M∩int□ spans M.
d)
dd!∣M∩int□∣≥dw−d.
e)
dd!volM(□)≥dw.
Proof.
The width is continuous with respect to the approximation limn→∞Conv(n1M∩□)=□.
Therefore we may suppose □ is the convex hull of finitely many points in MQ.
Let X be a toric
desingularization of the projective model of the graded ring ⊕n≥0⊕m∈M∩n□k⋅χm.
There exists a Q-Cartier divisor L on X, semiample and big, supported by the invariant prime divisors of X,
such that □ is the moment polytope of L. We compute
[TABLE]
Therefore the semi-invariant basis of Γ(⌈KX+L⌉) is in one-to-one correspondence to the interior
lattice points of □. Denote A=M∩int□. The linear system ∣⌈KX+L⌉∣ is non-empty if
and only if A is non-empty, it maps X onto a variety of the same dimension if and only if A−a generates the R-vector space
MR, for every a∈A, and maps X birationally onto a variety of the same dimension if and only if A−a generates the lattice
M, for every a∈A.
Let ϵ be the Seshadri constant of L at a very general point of X. We have
ϵ=ϵd(L). By Theorem 4.10, ϵ≥dw.
Therefore the claims are just a restatement of Theorem 5.1 for (X,L).
∎
Part a) of the theorem was proved by Kannan, Lovász with d2 replaced by cd2, for some constant c.
It is expected that the optimal bound is linear in d (see [12]). Parts b) and c) are new statements,
while d) and e) improve similar bounds in [12].
6. Appendix on Geometry of Numbers
We recall the definitions and results from the Geometry of Numbers that we use.
For proofs and more, the reader may consult [14].
Let M≃Zd be a lattice of rank d. The dual lattice N=Mˇ is defined
as HomZ(M,Z), and we have a duality pairing
N×M→Z,⟨φ,m⟩=φ(m).
A subset A⊆M spans the lattice M if the difference set A−A={a′−a;a′,a∈A} generates the lattice M.
The dimension of a subset A⊆MR, denoted dim(A), is the dimension of the R-vector space generated by A−A. The Q-dimension of a subset A⊆MR, denoted
dimQ(A), is the dimension of the Q-vector space generated by A∩MQ−A∩MQ.
We have dimQ(A)≤dim(A), and equality holds if A⊆MQ.
If □⊆MR is a convex set, then dimQ(□)=d if and only if dim(□)=d.
For a convex set □⊆MR, the polar convex set□∗⊆NR is defined as
[TABLE]
To □ we can associate the difference convex set □−□={m′−m;m′,m∈□}, which is [math]-symmetric.
The polar convex set (□−□)∗ consists of the linear functionals
φ∈NR such that the interval φ(□) has length at most 1.
The lattice width of a compact convex set□⊂MR is defined as the smallest
length of any interval φ(□), with all φ∈N∖0. It coincides with the
first minimum of Minkowski of (□−□)∗ with respect to N.
Let □⊆MR be a closed convex set which contains the origin. For i≥1,
the i-th successive minimum of (M,□) is defined by
[TABLE]
We obtain an increasing chain
0≤λ1≤λ2≤⋯≤λd+1=+∞.
Note that λi is a finite real number if and only if i≤dimQ(□), and
λi(M,c□)=λi(M,□)/c for every c>0. In case dim(□)=d, we
have the equivalent (Minkowski’s) definition
[TABLE]
The second main theorem of Minkowski, in the generalized form due to Davenport-Estermann,
is
Theorem 6.1**.**
Let □⊂MR be a convex set of dimension d. Then
[TABLE]
The transference theorem of Mahler, with the improved upper bound due to Banaszczyck [2, Theorem 2.1], is
Theorem 6.2**.**
Let □⊂MR be a [math]-symmetric compact convex set of dimension d.
Let □∗⊂NR be the polar body. Then
[TABLE]
With the original upper bound of Mahler ((d!)2 instead of d), the transference theorem
can be deduced from the second main theorem of Minkowski. The upper bound d is sharp
in dimension one. In dimension two, Banaszczyck [2, Proof of Theorem 2.1] mentions
the upper bound 2/3. We show next that the sharp upper bound in dimension two is in fact 3/2.
Sharp transference theorem in dimension two
Let □⊂R2 be a [math]-symmetric polytope, of dimension 2, with λ1(Z2,□)=1.
This means that Z2∩int□={0} and Z2∩∂□=∅.
We say that □ is maximal if every top face of □ contains a lattice point in its relative interior.
We bring □ into maximal position, preserving the initial hypothesis, as follows. Since □
is [math]-symmetric, the top faces come in pairs. Fix a pair F,−F. We slide out these top faces until
they either a) hit a lattice point in their relative interior, or b) these faces collapse to a point (they become
redundant). Apriori, it is also possible that we may slide out the faces to infinity, but one may easily
check that this is impossible. We repeat the argument for all pairs of top faces.
In finitely many steps, we enlarged □⊂□′ such that □′⊂R2 is a [math]-symmetric polytope,
of dimension 2, with λ1(Z2,□′)=1, and □′ is maximal. It is enough to prove transference for
□′.
From now on, we suppose □ is maximal.
Lemma 6.3**.**
On each top face F of □ choose exactly one relative interior lattice point uF, in such a way so that
−uF is the choice for the opposite face −F. Let Q be the convex hull of these lattice points. Then, after possibly
changing the basis of Z2, Q is one of the following:
The square with vertices (1,0),(0,1),(−1,0),(0,−1).
2)
The polytope with vertices (1,0),(0,1),(−1,1),(−1,0),(0,−1),(1,−1).
Proof.
Let u1,u2 be two adjacent vertices of Q. Then Conv(0,u1,u2)∖{u1,u2} is contained in
int□. Therefore Conv(0,u1,u2) contains no other lattice points besides its vertices. Therefore u1,u2
is a basis of Z2. We may suppose u1=(1,0) and u2=(0,1). Thus u1u2 is a top face of Q.
The opposite is also a top face. The other top faces are contained in the second and fourth quadrant. Enough to
look inside the second quadrant.
If Q has 4 vertices, we obtain 1). Suppose Q has at least 6 vertices. Then there are at least two top faces
in the second quadrant. But (0,1) and (−1,0) are vertices of Q, and (−1,1) is not contained in the interior of Q.
Therefore Q has exactly one vertex, namely (−1,1), in the interior of the second quadrant. We are in case 2).
∎
Case 1)
Here □=∩i=12{(x,y)∈R2;∣⟨ui,(x,y)⟩∣≤1}, where u1=(1,β1),
u2=(−β2,1) and β1,β2∈[0,1). Indeed, we may suppose (1,1) is separated from □ by the
edge passing through (1,0). The exterior normal to this edge is u1=(1,β1) for some β1≥0.
Since ⟨±u1,(0,1)⟩<1, we deduce β1<1. The lattice point (−1,1) must be separated from
□ by its edge through (0,1). The exterior normal to this edge is u2=(−β2,1) for some β2≥0.
Since ⟨±u1,(1,0)⟩<1, we deduce β1<1.
Conversely, any polytope □ as above satisfies Z2∩int□={0}.
Since u1,u2 are linearly independent, we compute
[TABLE]
Let z=(z1,z2)∈Zˇ2. Then
[TABLE]
and z∈h(z)⋅□∗, where
[TABLE]
We compute
[TABLE]
We claim that λ2(Zˇ2,□∗)≤23. Indeed, suppose 0≤β1≤β2<1. Then
[TABLE]
Therefore
[TABLE]
One may check that the right hand side is at most 23, with equality only for β1=0,β2=21.
Case 2)
Here □=∩i=13{(x,y)∈R2;∣⟨ui,(x,y)⟩∣≤1}, where u1=(1,β1),
u2=(β2,1), u3=(−β3,1−β3), and β1,β2,β3∈(0,1). Indeed, we may
suppose (1,1) is separated from □ by its edge through (1,0). Let u1=(1,β1) be the exterior normal,
so that β1≥0. We have ⟨u1,(0,1)⟩<1 and ⟨u1,(1,−1)⟩<1. That is 0<β1<1.
Let u2 and u3 be the normalized exterior normals to Q through (0,1) and (−1,1), respectively.
We have ⟨u2,(1,0)⟩<1 and ⟨u1,(−1,1)⟩<1, that is 0<β2<1. Similarly,
⟨u3,(0,1)⟩<1 and ⟨u3,(−1,0)⟩<1, that is 0<β3<1.
Conversely, any polytope □ as above satisfies Z2∩int□={0}. Indeed, once u1 is chosen,
it is enough to consider the lattice points in the strip ∣⟨u1,⋅⟩∣≤1. By symmetry, we may
take y>0. The lattice points in the strip with x≤−1 must be separated from □ by its top face through (−1,1).
We are left with the lattice points (0,n)(n=1,2,3,…). But these are separated from □ by its edge through (0,1).
Note that □∗ is the [math]-symmetric polytope with vertices ±u1,±u2,±u3, and is inscribed in the polytope
Q∗=Conv(±(1,0),±(1,1),±(0,1),±(−1,1)). For e∈Zˇ2∖0, let
h(e)=inf{t>0;e∈t⋅□∗}. Denote se=(1−h(e)−1)−1. We compute
[TABLE]
For t1,t2>0, we have t11+t21≥t1+t24. Thus
[TABLE]
Similarly, s1,0+s0,1>6 and s1,1+s0,1>6. Therefore two of the s1,0,s1,1,s0,1 are strictly larger than 3.
Therefore λ2(Zˇ2,□∗)<23. We obtained:
Theorem 6.4**.**
Let □⊂R2 be a [math]-symmetric convex body. Then
[TABLE]
Proof.
The first inequality is trivial. For the second, suppose by contradiction that it fails. Then we may approximate □ with a
polytope, and suppose □ is a polytope. If □ is a polytope, we may scale so that λ1=1, and we are
done by the two cases above.
∎
Example 6.5**.**
The upper bound 23 is attained for □ in Case 1) with β1=0,β2=21.
In this case, □=Conv(±(1,23),±(1,21)), □∗=Conv(±(1,0),±(−21,1)) and
λ1(Z2,□)=1,λ2(Zˇ2,□∗)=23.
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