Combinatorial Method in
Adjoint Linear Systems on Toric Varieties
H u i -We n L i n
1. Introduction
Inspired by minimal model theory, Fujita in the 1980s conjectured as follows.
Conjecture [6]. LetX be a nonsingular complex projective variety of dimen- sionn, and let D be an ample divisor on X. Then:
(I) KX+ D is generated by global sections for ≥ n + 1; and (II) KX+ D is very ample for ≥ n + 2.
Moreover, (I) and (II) should still hold true ifX has only “mild singularities”.
For nonsingular varieties, the one-dimensional case is an easy fact in curve the- ory. The two-dimensional case follows from the work of Reider [16]. In higher- dimensional cases, (I) is known forn = 3 [3] and n = 4 [8], and by [1] we know thatKX+12(n2+ n + 2)D is generated by global sections for all n. Less is known about (II) with one exception: ifD is already very ample, then (I) and (II) follow from Bertini’s theorem by induction on dimensions.
For part (I), allowingX to have rational Gorenstein singularities, Fujita him- self had shown (among other things) thatKX+ (n + 1)D is nef. For varieties over a field of arbitrary characteristic that have singularities ofF -rational type, Smith showed that (I) holds ifD is further assumed to be generated by global sections [17]. (In characteristic zero this can also be proved by using vanishing theorems.) Both [6] and [17] apply well to quite general toric varieties, since they have only rational singularities and on them a Cartier divisor is nef if and only if it is basepoint free (cf. Section 5). Moreover, ample divisors are automatically generated by global sections (Corollary 2.3). In fact, for nonsingular toric va- rieties, Fujita’s conjectures hold because ample divisors are automatically very ample (Demazure’s theorem).
These implications motivate our present work: results on toric varieties should admit direct proofs using only toric (combinatorial) techniques. In this note such elementary proofs are found for rather general toric varieties. Moreover, our com- binatorial treatment also provides results on the “very ampleness” conjecture (II).
Main Theorem. LetX be a complete toric variety of dimension n with ample (Cartier) divisorD.
Received May 10, 2002. Revision received September 9, 2002.
Supported by NSC project: NSC 90-2115-M008-020, Taiwan.
491
A The reflexive sheaf O(KX+ D) is generated by global sections for ≥ n +1.
IfX is Gorenstein, then KX+ nD is also generated by global sections unless (X, D) ∼= (Pn, O(1)).
B IfX is Gorenstein and Q-factorial, then KX+ D is very ample for ≥ n + 2 withn ≤ 6. For = n + 1 with n ≤ 4, this is also true unless (X, D) ∼= (Pn, O(1)). For n ≥ 7, KX+ D is very ample for ≥3
2n
− 1.
In Section 2 we review the necessary background in toric geometry. Section 3 contains elementary proofs of the main theorems A and B. An alternative toric proof (modeled on [6]) of Theorem A in the Gorenstein case is given in Section 4, where a toric proof of the singular version of toric Kodaira vanishing theorem is also given. (After completion of this work, I was informed that a proof recently appeared in a preprint by Mustata [13].)
We should remark here that a different toric proof of Theorem A in the Goren- stein case has been found by Laterveer [10] and Fujino [5] using Reid’s [15] toric version of Mori theory.
Acknowledgment. In an earlier version (authored with C.-L. Wang, dated Oc- tober 2000), it was claimed that Theorem B is true without the dimension restric- tionn ≤ 6 (cf. Remark 3.2). Upon finding this idea to be mistaken, Wang insisted on his removal as co-author. However, I remain grateful to him for many useful discussions while preparing this note.
2. Review of Toric Geometry
Only necessary material is recalled here, and readers are referred to [2; 7; 9; 14]
for details. LetN ∼= Znbe a lattice with dualM := HomZ(N, Z). A cone σ ⊂ NRwill mean a closed strongly rational polyhedral convex cone with dualσ∨ ⊂ MR defined by{u ∈ MR | u, v ≥ 0 ∀v ∈ σ}. Denote by ∂σ the collection of cones as faces ofσ. A fan of NR is a collection of cones{σ} such that (a) if τ ∈ ∂σ then τ ∈ and (b) if σ1, σ2∈ then σ1∩ σ2 is a face of bothσ1 and σ2. A p-dimensional cone is simplicial if it has exactly p edges. A fan is called complete if its cones fill upNR. In this paper, we consider mostly complete toric varieties; that is, is complete. We denote the subset of p-dimensional cones in
by p.
Fix a ground fieldk (or in fact we may take k = Z). For a cone σ ⊂ NR, we have thatSσ = σ∨∩ M determines a normal semigroup ring Aσ = k[Sσ] and an affine toric varietyUσ = Spec Aσ. The zero cone 0 ∈ corresponds to the com- mon Zariski open setU0 = Spec k[M ] ∼= Spec k[x1, x1−1, ..., xn, xn−1] ∼= T ∼= (k×)n. For a fan , X = X() is the toric variety defined by gluing all the Uσ. Here we associate to eachu ∈ M a monomial xu, so there is an obvious torus action ofT on X. For τ ∈ 1, we denote by ˆτ its (integral) primitive genera- tor. Defineτ⊥ := {u ∈ MR | u, ˆτ = 0}. This gives a codimension-1 subtorus Speck[τ⊥∩ M ] of T, and its closure in X in turn gives rise to a T (-invariant) Weil divisorDτ. On Uσ we have divxu =
τ∈1∩∂σ u, ˆτDτ. More generally, any w ∈ p gives rise to a(n − p) cycle.
Proposition 2.1. For aT -Weil divisor D =
aτDτ, the following statements hold.
1. (X, D) =
u∈PD∩Mk · xu, where PD = {u ∈ MR| u, ˆτ ≥ −aτ∀τ ∈ 1} is a convex but not necessarily integral polytope.
2. OnUσ, the reflexive sheaf O(D) corresponds to a finitely generated module Aσ xm(σ)1, ..., xm(σ)rσ over Aσ, where a minimal set of generators are as- sumed to be chosen; thenO(D) is generated by its global sections if and only if (iff )m(σ)j∈ PD for allσ and j.
AT (-invariant) Cartier divisor D is given by data (Uσ, xuσ) with σ ∈ n, uσ ∈ M, and uσ, ˆτ = uσ, ˆτ whenever τ ∈ 1∩ ∂σ ∩ ∂σ. The associated Weil di- visor is given by
τ∈1aτDτ, where aτ = uσ, ˆτ if τ ∈ 1∩ ∂σ. In this case,
(Uσ, D) = Aσ x−uσ and PD =
σ ∈n(−uσ+ σ∨). Note that T -Cartier divi- sors are in one-to-one correspondence with||-supported PL (piecewise linear) functions onNRthat areZ-valued onN. Namely, hD(v) = − uσ, v when v ∈ σ.
Let!D:X ··→ Ph0(X,D)−1be the rational map defined by the linear system|D|.
Proposition 2.2. LetD be a T -Cartier divisor. Then:
1. O(D) is generated by global sections iff !D is a morphism—that is,|D| is basepoint-free iffPD is an integral polytope with vertexes{−uσ | σ ∈ n} ( possibly with repetition) iff hD is convex;
2. D is very ample (that is, !D is a closed embedding) iff, for all σ ∈ n, uσ+ PD∩ M generates σ∨∩ M as a semigroup; and
3. D is ample (that is, D is very ample for large) iff PDis an integral polytope with vertexes{−uσ | σ ∈ n} (without repetition) iff hD is strictly convex.
Corollary 2.3. For complete toric varieties, ample divisors are generated by global sections. In fact,D is ample iff !Dis a finite morphism.
Toric varieties are naturally Cohen–Macaulay (they have only rational singulari- ties), with canonical(T -Weil) divisor K = −
τ∈1Dτ. Hence X is Gorenstein (resp.,Q-Gorenstein of indexr) iff K (resp., rK) is Cartier; that is, K is given by data{kσ∈ M | σ ∈ n} such that kσ, ˆτ = −1 (resp., −r) for τ ∈ 1∩ ∂σ. We also have thatX is Q-factorial iff is simplicial (i.e., consists of simplicial cones) and thatX is factorial iff the set of primitive generators of edges of each cone is part of aZ-basis ofM iff X is nonsingular.
Theorem 2.4. LetD be an ample divisor.
1. (Demazure) IfX is nonsingular then D is very ample.
2. (Ewald–Wessels [4]) For dimX = n ≥ 2, (n − 1)D is very ample.
3. Proof of the Main Theorem
We start with the following trivial but important observation. IfW = aτDτis a Weil divisor, then
IntPW∩ M = {u ∈ M | u, ˆτ + aτ > 0 ∀τ ∈ 1}
= {u ∈ M | u, ˆτ + aτ− 1 ≥ 0 ∀τ ∈ 1}
= PW+K ∩ M.
These equalities hold also for incomplete toric varieties (e.g.,Uσ).
Proof of Theorem A
Now letD be an ample divisor given by the local data (Uσ, xuσ). Applying the previous argument toW = D yields PD+K∩ M = Int PD∩ M.
For eachσ ∈ n, if we apply the foregoing to Uσ withW = 0 then we ob- tain the canonical module(Uσ, O(K)) = Aσ xm(σ)1, ..., xm(σ)r with exponents consisting of Intσ∨∩ M. Here we may (and do) choose {m(σ)j}j=1,...,r to be a minimal generating set that lies in the “quasi-box”
B(0,1]=
n
i=1
aivi | 0 < ai ≤ 1, v1, ..., vn:n distinct primitive
generators of (one-dimensional) edges of σ∨
(B[0,1]is defined similarly). Then{−uσ + m(σ)j}j=1,...,r is a minimal generat- ing set of(Uσ, O(D + K)). In order to show that (X, O(D + K)) generates O(D + K) on Uσ, we must show that PD+K contains the(−uσ+ m(σ)j)—in other words, thatuσ+ Int PD = Int((uσ+ PD)) contains the m(σ)j.
Let us define the “quasi-simplex”S[a,b]fora ≤ b to be the part of σ∨that has the form
aivi with 0 ≤ ai anda ≤
ai ≤ b, where the vi are among the primitive generators of edges ofσ∨(S(0,c), S(0,c], etc. are defined similarly). The point is that, sinceD is ample, c(uσ+ PD) ⊃ S[0,c]and Intc(uσ+ PD) ⊃ S(0,c)
for allc > 0. Hence
Int((n + 1)(uσ+ PD)) ⊃ S(0,n+1)⊃ B(0,1]⊃ {m(σ)j}j=1,...,r. This proves thatO((n + 1)D + K) is generated by global sections.
When X is Gorenstein, (Uσ, O(K)) = Aσ x−kσ with Int σ∨ ∩ M =
−kσ+ σ∨∩ M. For nD + K to be generated by global sections is equivalent to having, for allσ ∈ n, that −kσ∈ Int n(uσ+ PD). However, Int(n(uσ+ PD)) ⊃ S(0,n)⊃ B(0,1)andB(0,1] −kσand so it follows that, if there is aσ with −kσ /∈
Intn(uσ + PD), then −kσ must lie in the boundary of n(uσ + PD) and be of the form−kσ = v1+ · · · + vn, where the vi are distinct primitive generators of σ∨. Moreover, there are no lattice points in Int B[0,1)(actually, no lattice points in Intn(uσ+ PD)) nor any lattice points in lower faces of B[0,1], except the vertices.
This implies thatv1, ..., vnform aZ-basis ofM.
We claim that there is no other primitive generatorv of σ. If such a v did exist, then IntB[0,1)∩ M = ∅ implies that any n −1 vi, together with v, would still form aZ-basis ofM. Write v = n
i=1civi. Computation of determinant shows that
|ci| = 1 for all i. Since v is also a primitive generator, at least one cimust be−1.
LetI be the subindex set of {1, ..., n} such that ci = −1 for all i ∈ I and cj = 1
for allj /∈ I. Consider the element w = v +
i∈Ivi =
j /∈Ivj. Then w =
1 2
v +n
i=1vi
would be a nontrivial interior lattice point of Intn(uσ+ PD), a contradiction.
It follows that, ifnD + K is not generated by global sections on some Uσ, thenn(uσ + PD) = S[0,n]and the polytope uσ + PD = S[0,1] is the regularn- simplex, withv1, ..., vnthe edges through 0. Because D is ample, this implies that(X, D) ∼= (Pn, O(1)).
Proof of Theorem B
TheQ-factorial assumption asserts that all cones involved are simplicial. Let us first suppose thatX is singular on some open set Uσ withσ ∈ n. Since X is Gorenstein, this is equivalent (by our previous argument) to−kσ = v1+ · · · + vn
forvithe primitive generators of edges ofσ∨. Claim 3.1. If ≥ max
n + 1,3
2n
− 1
, then D + K is very ample on Uσ. That is,!D+K|Uσ is a closed embedding.
Proof. LetB[0,1], S(a,b), S[a,b), ... be the same subsets of σ∨as before. We set also Sc= S[c,c]. There is an obvious reflection of lattice points in B[0,1]with respect to the center 12
vi, namely B[0,1]∩ M α → α=
vi − α ∈ B[0,1]∩ M. This reflectsS[a,b)toS(n−b,n−a].
By assumption we have that−kσ∈ Sλfor 0< λ < n (λ ∈ Q) and there is no interior lattice point inS[0,λ). By reflection, there is also no interior lattice point inS(n−λ,n]∩ B[0,1].
If−kσ ∈ Int B[0,1] thenn − λ ≥ λ (i.e., λ ≤ n/2). In general, let −kσ =
n
i=1aivi. By reordering the viif necessary, we may assume that there exists an m ∈ N with 0 < ai < 1 for i ≤ m and with ai = 1 for i ≥ m + 1. If m < n then−kσ is in the interior of an “upper face”Fm ofB[0,1]. In this case −k :=
m
i=1aiviis an interior lattice point of the “lower face”FmofB[0,1], which makes the coneσ∨ spanned byv1, ..., vmGorenstein because{vm+1, ..., vn} and −k generate−kσ:
m ∈ Int σ∨∩ M ⇒ m + vm+1+ · · · + vn∈ Int σ∨∩ M
⇒ m + vm+1+ · · · + vn= −kσ+ γ, γ ∈ σ∨∩ M
⇒ m = −k+ γ (and so γ ∈ σ∨∩ M ).
Observe also that whenm < n there are no interior lattice points of B[0,1]and {vm+1, ..., vn} is a Z-basis of R vm+1, ..., vn ∩ M. Moreover, B[0,1]∩ M is gen- erated by{vm+1, ..., vn} ⊂ S1 andFm∩ M, on which −k∈ Sλ withλ≤ m/2 andλ = λ+ (n − m) ≤ n − m/2.
Letβ ∈ B[0,1]∩ M. If β ∈ Int σ∨, the Gorenstein property implies that β =
−kσ+ γ for some γ ∈ σ∨∩ M and γ ∈ S[0,n−λ]. If β /∈ Int σ∨, then we may as- sume moduloS1thatβ ∈ Int Fm(form = n, Fm:= B[0,1]). The result of Ewald and Wessels [4] states thatβ is generated by S[0,m−1]∩Fm∩M (cf. Theorem 2.4(2) and Remark 3.2).
The “very ampleness” of D + K on Uσ is equivalent to the fact that (uσ+ kσ) + PD+K∩ M generates B[0,1]∩ M, since the latter generates σ∨∩ M via translations. Given
(uσ+ kσ) + PD+K ∩ M = kσ+ uσ+ Int PD∩ M
= kσ+ Int((uσ+ PD)) ∩ M
⊃ kσ+ S(0,)∩ M ⊃ S[0,−λ)∩ M,
we need only require that − λ > max{n − λ, m − 1, λ,1}. That is, >
max{n, λ + m − 1, λ + λ, λ + 1}. Now
λ + m − 1 < n + m/2 − 1 ≤ 32n − 1, λ + λ= 2λ+ (n − m) ≤ n,
λ + 1 < n − m/2 + 1 ≤ n + 1/2.
The claim is proved.
Notice that3
2n
− 1 ≤ n + 2 for n ≤ 6 and3
2n
− 1 ≤ n + 1 for n ≤ 4. In the rangen ≤ 4, KX+ (n + 1)L fails to be very ample only if X is nonsingular on someUσ, hence −kσ =
vi∈ Sn. Since the vialready form aZ-basis ofM and (n + 1)uσ+ kσ+ P(n+1)D+K∩ M ⊃ kσ+ S(0,n+1)∩ M ⊃ S[0,1)∩ M, it is clear that(uσ + kσ) + PD+K ∩ M ⊃ S[0,1]∩ M for all > n + 1 and that
D + K is very ample on Uσfor ≥ n + 2. (In particular, this gives a simple toric proof of Fujita’s conjecture for nonsingular toric varieties in any dimensions.)
Moreover, if(n + 1)D + K fails to be very ample on Uσ, then (n + 1)uσ + kσ+ P(n+1)D+K∩ M ⊃ S[0,1]∩ M. That is, vi /∈ (n +1)uσ+ kσ+ P(n+1)D+K∩ M for some (in fact, all)i. This implies that (n + 1)(uσ+ PD) = S[0,n+1]. Indeed, if (n + 1)(uσ+ PD) properly contains S[0,n+1]then
vi+ (−kσ) = vi+ (v1+ v2+ · · · + vn) ∈ Sn+1∩ Int σ∨∩ M
⊂ Int(n + 1)(uσ+ PD) ∩ M = (n + 1)uσ+ P(n+1)D+K∩ M, which is a contradiction!
Therefore, the polytopeuσ+ PDmust be the regularn-simplex with v1, ..., vn the edges through 0. SinceD is ample, this implies that (X, D) ∼= (Pn, O(1)).
Remark 3.2. Wang has conjectured that, for ann-dimensional Gorenstein cone σ∨with−kσ∈ Int B[0,1], B[0,1]is generated byS[0,n/2]∩ M. If this is true then the foregoing argument will lead to a proof of conjecture (II) in the singular case in any dimension. No counterexample has been found in a Maple program search.
4. Toric Vanishing Theorems
The following Kodaira-type vanishing theorem (Theorem 4.1(2)) for ample line bundles on toric varieties was stated without proof in [2, (7.5.2)] and [14, p. 130].
For the reader’s convenience we give a proof here assuming only that the bundle is big and nef (Kawamata–Viehweg vanishing theorem). Part 1 is a more standard fact in toric geometry. We put them together not only for completeness but also because their proofs are along the same line.
Theorem 4.1. LetX be a complete toric variety, and let D be a Cartier divisor that is generated by global sections.
1. Hi(X, D) = 0 for i ≥ 1.
2. If, moreover,D is big (e.g., ample), then Hi(X, K + D) = 0 for i ≥ 1.
Proof. By Demazure’s graded decomposition theorem for the Cartier divisorL with associated PL functionh (see [2, 7.2; 9, p. 42]),
Hi(X, L) =
m∈M
HZ(mi ,h)(NR, k),
whereZ(m, h) = {n ∈ NR| m, n ≥ h(n)}. Moreover, for i ≥ 2 we have Hi−1(NR− Z(m, h), k) ∼= HZ(mi ,h)(NR, k),
and then there exists an exact sequence
0→ HZ(m0 ,h)(NR, k) → k → H0(NR− Z(m, h), k) → HZ(m1 ,h)(NR, k) → 0.
Note thatZ(m, h) = NRif and only ifxmis a section ofL.
For the proof of part 1, letL = D. Since h is convex (D is generated by global sections),NR− Z(m, h) = {n ∈ NR | m, n < h (n)}, which is a convex open cone (hence, contractible) and soHZ(mi ,h)(NR, k) = 0 for i ≥ 2. To achieve the desired vanishing fori = 1, if xm is a section ofD then Z(m, h) = NR and H0(NR− Z(m, h), k) = H0(∅, k) = 0, so HZ(m1 ,h)(NR, k) = 0. And if xmis not a section ofD then Z(m, h) = NR. By the definition of local cohomology, we haveHZ(m0 ,h)(NR, k) = 0. The previously displayed exact sequence again implies thatHZ(m1 ,h)(NR, k) = 0.
Part 2, by a Grothendieck–Serre duality theorem for Cohen–Macaulay schemes (see [2, (7.7.1)] for a toric proof in the toric case), is equivalent toHi(X, −D) = 0 for alli ≤ n − 1. Let L = −D. Then h = h−D= −hDis concave, soZ(m, h) is a closed convex cone. In this case−D has no sections, so Z(m, h) = NR for allm. By an argument similar to our proof of part 1, we have HZ(m0 ,h)(NR, k) = 0 andHZ(m1 ,h)(NR, k) = 0.
For other i, since D is assumed to be big, it follows by Lemma 4.2 that Z(m, h) cannot contain any positive-dimensional vector subspace of NR: if for somen = 0 we have n ∈ Z(m, h) and −n ∈ Z(m, h), then by adding together hD(n) + m, n ≥ 0 and hD(−n) + m, −n ≥ 0 we obtain (by convexity of hD) that 0= hD(n + (−n)) ≥ hD(n) + hD(−n) ≥ 0. That is, hD(−n) = −hD(n) and sohD|Rn is linear—a contradiction. Notice thatZ(m, h) may consist of just a single point 0. Now it is easy to see thatHj(NR− Z(m, h), k) = 0 for 1 ≤ j ≤ n − 2, since NR− Z(m, h) is either contractible or homotopic equivalent to the (n − 1)-dimensional unit sphere. The proof is complete.
Lemma 4.2. LetD be a Cartier divisor that is generated by global sections. Let d be the dimension of the maximal vector subspace V of NR such thathD|V is linear, and let!D be the projective morphism defined by|D|. Then
dim Image!D= dim PD = n − d.
Proof. The first equality follows from basic properties of Kodaira dimension. For the second equality, recall thatPD= {u ∈ MR| u ≥ hD} by Proposition 2.1(1) and the definition ofhD. If the vi are a basis ofV, then u ∈ PDimplies thatu(vi) ≥ hD(vi) and −u(vi) = u(−vi) ≥ hD(−vi) = −hD(vi). That is, u|V = hD|V. Hence the degree of freedom ofu is n − d.
One also has a nice understanding ofH0(X, K + D) by the following lemma.
Lemma 4.3 [7, p. 90]. LetX be a complete Gorenstein toric variety and D an ample (Cartier) divisor. If (X, K + D) = 0 then K + D is generated by global sections. In fact,PK+Dis the convex hull of IntPD∩ M.
Example 4.4. The conclusion of Lemma 4.3 is wrong ifX is only Q-Gorenstein.
LetM ∼= Z4 withv1 = (1, 0, 0, 0), v2 = (0,1, 0, 0), v3 = (0, 0,1, 0), v4 = (1,1,1, 3), and P the convex hull of 0, v1, v2, v3, v4. Now P determines a Q- factorial toric varietyX() and an ample divisor D such that PD = P. Let σ∨be the cone spanned byv1, v2, v3, v4(and henceuσ = 0). Then the canonical mod- ule(Uσ, O(K)) = Aσ xm1, xm2, with m1= (1,1,1,1) and m2= (1,1,1, 2); in fact, IntB[0,1]∩M = {m1, m2}. It is easily seen that P2D+K∩M = Int P2D∩M = Int 2P ∩ M, which contains m2 but notm1. So (X, K + 2D) = 0, but the re- flexive sheafO(K + 2D) is not generated by its global sections on Uσ.
This example is inspired by the work of Ewald and Wessels [4]. It can easily be generalized to higher dimensions.
Alternative Proof of Theorem A in the Gorenstein Case
By Lemma 4.3 we need only show thatK + D has a nontrivial section for some
≤ n +1. The Euler characteristic p() := χ(X, K + D) is a polynomial in of degree≤ n and in the range ∈ N, p() = h0(X, K + D), by Theorem 4.1(2).
IfK + D has no sections for 1 ≤ ≤ n, then p() has roots 1, ..., n and hence p(n + 1) = 0, because p is a nontrivial polynomial.
IfK + nD is not generated by global sections, then p() has roots 1, ..., n.
Therefore,
p() = χ(X, K + D) = c( − 1) · · · ( − n).
Using the formula given by the Riemann–Roch theorem for line bundles on pos- sibly singular toric varieties,
p() = (−1)nχ(X, −D) =
e−D·
1−K
2 + · · ·
(n)
= Dn
n! n+ Dn−1K
2(n − 1)!n−1+ O(n−1)
(see [7, Sec. 5.3]), we get thatc = Dn/n! and Dn−1K = −(n + 1)Dn. That is, (K + (n + 1)D) · Dn−1= 0.
BecauseK +(n+1)D is effective and D is ample, this implies that K +(n+1)D = 0. But then Dn = p(n + 1) = h0(X, K + (n + 1)D) = h0(X, O) = 1, so h0(X, D) = h0(X, K + (n + 2)D) = p(n + 2) = n + 1. Consider the projective morphism!D:X → Pndefined by|D| with D = φ∗H, where H is the hyper- plane class. It is, in general, a finite morphism by Corollary 2.3. Moreover,!Dis also a birational morphism (of degree 1) ontoPnsinceDn= 1. Hence !D is an isomorphism and(X, D) ∼= (Pn, O(1)).
Remark 4.5. The idea of this proof follows Fujita’s paper [6] closely. It uses the Riemann–Roch theorem and so is not as elementary as the previous proof in Sec- tion 3. My motivation for giving this proof is to demonstrate a special feature of toric varieties.
Remark 4.6. TheoremA in the Gorenstein case has been proved by Laterveer [10]
using different methods. In [10] it is also claimed thatK +(n+2)D is very ample.
However, there is a mistake in [10, p. 457]: If we replacet, L, and X by n + 2, O(1), and Pn(respectively) then we get a contradiction to his claim that the ratio- nal polyhedronPKX+tL= PLcontains the rational polyhedronP(t−1)L= P(n+1)L. The correct version of this inclusion isPKX+tL∩ M = Int PtL∩ M.
Question 4.7. In the second part of Theorem A, can one relax the assumption onX to be Q-Gorenstein or perhaps even all the assumptions? Also, can one re- move theQ-factoriality assumption onX in Theorem B?
5. Appendix: Toric Nakai–Moishezon–Kleiman Criterion Results in this section are well known to experts and are essentially contained in [14; 15], though not stated in generality here. Because they are crucial for us to fix ideas when working on toric varieties, we give the proofs for the reader’s conve- nience. (In fact, the result in this appendix has already appeared in [12]; however, it is hoped that the treatment here has some independent interest.)
Assume first that is a complete simplicial fan of dimension n and that D is aT -invariant Cartier divisor with data (Uσ, xuσ). Let ω ∈ n−1and letlωbe the corresponding 1-cycle as in Section 2. Suppose thatω separates two cones σ and σinn. Let e1, ..., en−1be the primitive generators of edges ofω, and let enand en+1be the primitive generators of opposite edges ofσ and σ, respectively. Be- causee1, ..., enform aQ-basis ofN, we have the relationn+1
i=1aiei = 0 with an+1= 1 and an> 0. Recall now the following formulas from [15, (2.7)]:
(1) Delω= 0 if e /∈ {e1, ..., en+1},
(2) Deilω= aiDen+1lωfori = 1, ..., n, and (3) Den+1lω= mult(ω) mult(σ) > 0,
where mult(ω) = [Nω:Ze1+ · · · + Zen−1] andNωis the sublattice ofN gener- ated (as a group) byω ∩ N, and similarly for mult(σ) (see [7, p. 100]).
Lemma 5.1. Dlω= uσ− uσ, en+1Den+1lω. Proof. By formula (1),Dlω=n+1
i=1deiDeilω, which equals n+1
i=1deiai Den+1lω by (2). Fori = 1, ..., n we have dei= uσ, ei, so
n+1 i=1
aidei=
uσ,n
i=1
aiei
+ den+1an+1
= uσ, −en+1 + uσ, en+1 = uσ− uσ, en+1 and thenDlω= uσ− uσ, en+1Den+1lω.
Proposition 5.2. LethD be the PL function defined byD. Then 1. hDis convex onσ ∪ σiff Dlω≥ 0, and
2. hDis strictly convex onσ ∪ σiff Dlω> 0.
Proof. Notice thathD is convex iffhD(w) ≤ − uσ, w for all w ∈ 1andσ ∈
n. That is, uσ− uσ, w ≥ 0 for all w ∈ σ. By Lemma 5.1 and formula (3), this is equivalent toDlw≥ 0. The strictly convex case is entirely similar.
Theorem 5.3 (Toric Nakai–Moishezon–Kleiman criterion). For any complete toric varietyX with D a Cartier divisor:
1. D is generated by global sections iff D is nef ; and 2. D is ample iff D is numerically positive.
Proof. If the fan is simplicial then this follows from Proposition 2.2 and Proposi- tion 5.2. In the general case, part 1 again follows from the simplicial case: consider subdivision of into the simplicial fan and letφ : X= X() → X = X() be the corresponding toric birational morphism. Then notice thatD is nef on X iffφ∗D is nef on Xand thatφ∗D is generated by global sections on XiffD is generated by global sections onX.
Part 2 follows from part 1: D is ample certainly implies that it has positive de- gree when restricted to any effective curve; conversely, ifD is numerically positive then by part 1|D| defines a morphism !D, which has no positive-dimensional fiber because otherwiseD would have zero degree along curves in the fiber. Hence !D is finite, and this implies thatD is ample.
Added in proof. Sam Payne has informed the author that Lemma 4.3 quoted from [7], on which our alternative proof of Theorem A is based, does not seem to have a known valid proof.
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Department of Mathematics National Central University Chung-Li
Taiwan
linhw@math.ncu.edu.tw
