Classifying smooth lattice
via toric fibrations
We show that any smooth -normal lattice polytope of dimension and degree is a strict Cayley polytope if . This gives a sharp answer, for this class of polytopes, to a question raised by V. V. Batyrev and B. Nill.
Let be an -dimensional lattice polytope (i.e., a convex polytope with integer vertices) in . We represent it as an intersection of half spaces
where are the half spaces, the supporting hyperplanes, the corresponding primitive inner normals, and . Recall that an -dimensional lattice polytope is smooth if every vertex is equal to the intersection of of the hyperplanes , and if the corresponding normal vectors form a lattice basis for . Smooth polytopes are sometimes called Delzant polytopes or regular polytopes.
Let be an -dimensional lattice polytope. Define for .
Note that the lattice points of are precisely the interior lattice points of .
Let be a smooth -dimensional lattice polytope and an integer. Let be a vertex of . Reorder the hyperplanes so that . We say that is -spanned at if the lattice point , defined by , lies in . We say that is -spanned if is -spanned at every vertex.
If , we can write in the dual basis of , and similarly . It follows from the definition that if is -spanned, then .
Let be the smooth polytope obtained from the simplex by removing the simplex , see Figure 1. Assume . Then , but is not -spanned. In fact, consider the vertex of , given by , then the lattice point , given by , is not a point in . Similarly for the vertices and .
Note that if we instead remove , then the resulting polytope is -spanned. In this case, the vertices , , and all “go” to the same lattice point , which is an interior point of the polytope.
We recall the definitions of degree and codegree of a lattice polytope introduced in .
Let be an -dimensional lattice polytope. The codegree of is the natural number
The degree of is
We further introduce a more “refined” notion of codegree.
Let be an -dimensional lattice polytope. The -codegree of is defined as
The number is well defined, since and, for any polytope , we have if and only if for every integer . Moreover, it is clear that holds.
Let denote the -dimensional simplex. Then we have and .
As we shall see in Proposition 2.2, the following definition embodies the polytope version of the notion of nef value for projective varieties.
Let be an -dimensional smooth lattice polytope. The nef value of is
Clearly, the inequality always holds. It can be strict, as in Example 1.3, where and .
Observe also that when is an integer, it follows from Lemma 2.4 below that .
An -dimensional lattice polytope in is called -normal if .
We shall now explain the notion of generalized Cayley polytopes — these are particular examples of the twisted Cayley polytopes defined in .
Let be lattice polytopes in , a basis for and . If are the vertices of , so that is the convex hull, and is a positive integer, consider the polytope . Any polytope which is affinely equivalent to this polytope will be called an th order generalized Cayley polytope associated to , and it will be denoted by
If all the polytopes have the same normal fan (equivalently, if they are strictly combinatorially equivalent), we call strict, and denote it by
If in addition , we write and call a strict Cayley polytope.
Smooth generalized strict Cayley polytopes are natural examples of -normal polytopes. In Proposition 3.9 we give sufficient conditions for to be -normal, and we compute the common value in this case.
In  Batyrev and Nill posed the following question:
Question. Given an integer , does there exist an integer such that every lattice polytope of degree and dimension is a Cayley polytope?
A first general answer was given by C. Haase, B. Nill, and S. Payne in , where they prove the existence of a lower bound which is quadratic in .
In this article we give an optimal linear bound in the case of smooth -normal lattice polytopes, and show that polytopes of dimension greater than or equal to this bound are strict Cayley polytopes.
Answer. For -dimensional smooth -normal lattice polytope, we can take . More precisely, if is a smooth -normal lattice polytope of dimension and degree such that , then is a strict Cayley polytope.
Observe that the condition is equivalent to . Furthermore, note that our bound is sharp: consider the standard -dimensional simplex and let as in Example 1.6. Then, and is not a Cayley polytope. It is, however, a generalized Cayley polytope, with . We conjecture that an even smaller linear bound, like , should suffice for the polytope to be an -th order generalized Cayley polytope.
We shall deduce our answer from Theorem 1.12 below, which gives a characterization of -normal smooth lattice polytopes with big codegree. Before stating our main theorem, we recall the notion of a defective projective variety.
Let be a projective variety over an algebraically closed field and denote by its dual variety in the dual projective space. Then is defective if is not a hypersurface, and its defect is the natural number .
Let be a smooth lattice polytope of dimension . Then the following statements are equivalent:
is -normal and ,
is a smooth strict Cayley polytope, where and ,
the (complex) toric polarized variety corresponding to is defective, with defect .
We conjecture that the assumption always holds for smooth lattice polytopes satisfying and we therefore expect that the above classification holds for all smooth polytopes.
Our proof of Theorem 1.12 relies on the study of the nef value of nonsingular toric polarized varieties. This is developed in Section 2. Section 3 contains the study of generalized Cayley polytopes in terms of fibrations. In particular, for smooth generalized strict Cayley polytopes such that and for all , Proposition 3.9 shows that , providing a family of lattice polytopes without interior lattice points. Section 4 contains the proof of the classification Theorem 1.12, together with some final comments.
2. The codegree and the nef value
Let be a nonsingular projective variety over an algebraically closed field, and let be an ample line bundle (or divisor) on .
Assume that the canonical divisor is not nef. The nef value of is defined as
By Kawamata’s rationality theorem [9, Prop. 3.1, p. 619], the nef value is a positive rational number.
Let be a nonsingular projective toric variety of dimension , and let be an ample line bundle on . Let be the associated -dimensional smooth lattice polytope. Then
If we write the polytope as , then , where the are the invariant divisors on . Since , the polytope associated to and the adjoint line bundle is
The lattice points of form a basis for the vector space of global sections of ,
(see [13, Lemma 2.3, p. 72]).
Denote by the fan of , let be the fixed point associated to the -dimensional cone , and let be the affine patch containing . The restriction of a generator to is
where is a system of local coordinates such that It follows that the line bundle is spanned, or globally generated, at (i.e., it has at least one nonvanishing section at ) if and only if the lattice point , written with respect to the dual basis of , is in .
Because the base locus of a line bundle is invariant under the torus action, if it is non empty, it must be the union of invariant subspaces. Hence it has to contain fixed points. It follows that is spanned if and only if it is spanned at each fixed point, hence if and only if the polytope is -spanned. ∎
Let be an -dimensional smooth lattice polytope. Then
if and only if
Let be the polarized nonsingular toric variety corresponding to . In [12, Cor. 4.2] it is proven that is spanned for any line bundle on such that for all invariant curves on , unless and
Because the line bundle is ample, we have and thus , for all invariant curves . It follows that if or , then . This proves . If , then , because . The corresponding polytope is and , as stated in . ∎
Let be a smooth -dimensional polytope with codegree and nef value . Then and .
Let be the polarized projective toric variety associated to . Then . By [2, Lemma 0.8.3] is nef (and not ample). For any , is also nef. When is an integer, this implies that is spanned, hence . Taking and observing that has no lattice points, we deduce that , as claimed. ∎
Let where are coprime. On complete toric varieties, nef line bundles with integer coefficients are spanned (see [12, Thm. 3.1]). It follows that the linear system defines a morphism
where . The Remmert–Stein factorization gives where is a morphism with connected fibers onto a normal toric variety such that and . Moreover, is the contraction of a face of the nef cone NE() [3, Lemma 4.2.13, p. 94].
If the morphism contracts a line, i.e., if there is a curve such that and is a point, then is necessarily an integer. In fact implies .
Let be a smooth -dimensional polytope and let be the corresponding polarized toric variety. If , then there exists an invariant line on contracted by the nef value morphism . In particular, , and . If, moreover, is not birational, then is the contraction of an extremal ray in the nef cone, unless is even and .
Because the nef value morphism is the contraction of a face of the Mori cone, it contracts at least one extremal ray. Take to be a generator of this ray. Recall that, by Mori’s Cone theorem (see e.g. [6, p. 25]), . Because , we have
We will also need the following key lemma. This lemma, and its proof, is essentially the same as [3, Lemma 7.1.6, p. 157].
Let be the polarized nonsingular toric variety associated to a smooth -normal lattice polytope . Then the morphism is not birational.
Let Assume the morphism is birational. By [3, Lemma 2.5.5, p. 60] there is an integer and an effective divisor on such that
It follows that , and thus , which contradicts the assumption that is -normal. ∎
3. Generalized Cayley Polytopes and toric fibrations
Strict generalized Cayley polytopes (recall Definition 1.10) correspond to particularly nice toric fibrations, namely projective bundles. We will compute their associated nef values and codegrees. We refer to [5, Section 3] for further details on toric fibrations.
A polarized toric fibration is a quintuple where
and are normal toric varieties with ,
is an equivariant flat surjective morphism with connected fibers,
the general fiber of is isomorphic to the (necessarily toric) variety ,
is an ample line bundle on .
There is a - correspondence between polarized toric fibrations and fibrations of polytopes, making Definition 3.1 equivalent to the following.
Let be a surjective map of lattices and let be lattice polytopes. We call a fibration with fiber if
for every ,
are all distinct and are the vertices of
have the same normal fan, .
In [5, Lemma 3.6] it is proven that is a toric fibration if and only if the polytope associated to has the structure of a fibration. More precisely, is a toric fibration if and only if there is a sublattice such that the dual map is a fibration of polytopes with fiber . Moreover, is the toric variety defined by the inner normal fan of , and every fiber of , with the reduced scheme structure, is isomorphic to .
Observe that by construction, , where . The projection
gives the polytope the structure of a a fibration with fiber .
It follows that defines a toric fibration with general fiber isomorphic to .
The strict Cayley sum is associated to the toric fibration , where is the tautological line bundle and is the projection (see Figure 2).
Even if all the are smooth and the fiber is smooth, the polytope is not necessarily smooth. Consider the polytope depicted in Figure 3. At the vertex , the first lattice points on the corresponding three edges give the vectors , , and , which do not form a basis for the lattice.
When the polytopes are not strictly combinatorially equivalent, the variety associated to the Cayley polytope is birationally equivalent to a toric variety associated to a strict Cayley polytope, in the following precise way.
The normal fan defined by the Minkowski sum is a common refinement of the normal fans defined by the polytopes (see e.g. [15, 7.12]). Let be the polarized toric variety associated to the polytope and let be the induced birational morphism, where is the toric variety defined by the fan . Notice that the line bundle on is spanned, and the associated polytope is affinely equivalent to .
Set for . Note that the are strictly combinatorially equivalent, since their inner normal fan is . The normal fan of the polytope is then a refinement of the normal fan of the polytope , and thus it defines a proper birational morphism
Let be strictly combinatorially equivalent polytopes such that the polytope Ê is smooth. Let be the associated toric fibration. Then is smooth for all , and all fibers are reduced and isomorphic to .
Let be an invariant fiber of , then, since it is not general, it must be the fiber over a fixed point of . Equivalently, there is a vertex of which is the intersection of a -dimensional face such that and an -dimensional face such that . Hence . Moreover, by construction, is a simplex (possibly non standard) of edge lengths , with . Because is smooth, the first lattice points on the edges meeting at form a lattice basis. This is equivalent to asking that the matrix formed by taking the integral vectors as columns, has determinant . After reordering we can assume that . Then the corresponding matrix has the shape featured in Figure 4,
where , , corresponds to the coordinate of the first lattice point on the th edge of the simplex . The matrix is the matrix given by the first lattice points through the (corresponding) vertex of . A standard computation in linear algebra gives . It follows that and .
Because each vertex of is the intersection of a smooth fiber with , we conclude that is smooth for all . The equalities show that all invariant fibers have edge length . Every special fiber (as a cycle) is a combination of invariant fibers. If there were a non reduced fiber , then it should contain an invariant curve such that is numerically equivalent to , where is an invariant curve in . Then, because and , we would have . Lemma 3.6 implies that the are smooth, and hence is smooth. One can see this also from the standard fact that for every morphism with connected fibers between two normal toric varieties, the general fiber is necessarily toric [8, Lemma 1.2]. ∎
Observe that the hypothesis that is smooth in Lemma 3.6 above, is essential in order to prove that all fibers are reduced and embedded as Veronese varieties.
Toric projective bundles are isomorphic to projectivized bundles of a vector bundle, which necessarily splits as a sum of line bundles [8, Lemma 1.1]. We will denote the projectivized bundle by , where is the toric variety associated to .
Let be strictly combinatorially equivalent polytopes such that
is smooth. Then there are line bundles on such that the toric variety , defined by the inner normal fan of , is isomorphic to
Denote by the ample line bundle on associated to the given polytope . By Lemma 3.6, all fibers are isomorphic to and thus has the structure of a projective bundle over . Equivalently, and , where . ∎
Throughout the rest of the section we will always assume that is a smooth polytope. Denote as before by the associated polarized toric variety. Let be as above. The invariant curves on are of two types:
pullbacks of invariant curves from , corresponding to the edges of ,
curves contained in a fiber , corresponding to the edges on simplices .
Line bundles on can be written as , where is a line bundle on and is the tautological line bundle on
Because a line bundle on a nonsingular toric variety is spanned, respectively (very) ample, if and only if the intersection with all the invariant curves is non negative, respectively positive, there is a well understood spannedness and ampleness criterion, see [7, Prop. 2].
Let be a line bundle on . Then for every curve of type , we have , and for every curve we have . Consequently
is ample if and only if and is ample for all ,
is spanned if and only if and are spanned for all .
Because the fibers of correspond to simplices and thus are embedded as -Veronese varieties, and because the line bundles are ample, we see that corresponds to the toric embedding
Let be a smooth generalized strict Cayley polytope. Assume that for all . Then is -normal, and
where is ample on , for , and is the polytope defined by the line bundle , where .
Recall also that the canonical line bundle on is , where . It follows that
By Lemma 3.8, is spanned (resp. ample) if and only if (resp. ) and is spanned (resp. ample).
Observe that, because is ample for each , is ample if [12, Cor. 4.2 (ii)], which holds by assumption. It follows that, with the given hypotheses, we have
is spanned if and only if
is ample if and only if ,
and thus .
Consider the projection map such that . Clearly, if are such that , then for all points , . This implies that , and hence . Together with the inequality , this proves
Christian Haase showed us the following beautiful geometric argument to prove Proposition 3.9. As before, denote by the common normal fan of and by the normal fan of . Consider the smooth toric varieties and , and let denote the ample line bundle on with associated polytope and the ample line bundles on with associated polytopes .
As , we have that an integer multiple of is ample on for all . Let be the primitive generators of the one dimensional cones in , so that
Then the polytope
corresponding to the line bundle (with coefficients in ) , is combinatorially equivalent to , for all . On the other hand, the polytope
equals the barycenter of the simplex . Hence the polytope associated to reduces to the fiber over in , which can be identified with , and no multiple of the corresponding line bundle is ample.
For any rational number , we have again that the polytope
is combinatorially equivalent to for all . But now the polytope given by the points in at lattice distance from each of its facets is also combinatorially equivalent to , and therefore the polytope corresponding to is combinatorially equivalent to . We conclude that , as wanted.
4. Classifying smooth lattice polytopes with high codegree
We now use the results in the previous sections to give the proof of Theorem 1.12.
Proof of Theorem 1.12. Let be the nonsingular toric variety and ample line bundle associated to .
Assume (1) holds. Because is -normal, Lemma 2.7 implies that the nef value map is not birational. Set . Lemma 2.4 gives that . If is even and , then . Therefore, Lemma 2.6 implies that is an integer and , and that is a (non birational) contraction of an extremal ray in the nef cone of .
By [14, Cor. 2.5, p. 404], we know that is flat, is a smooth toric variety, and, since is smooth, a general fiber is isomorphic to , where . Under this isomorphism, , for some positive integer . Let be a line. Then, since , and hence , is contracted by , we have
We therefore get
which gives and . By Lemma 2.4, we have . As , we get . Hence , so that .
Since for a general fiber of and is flat, for every fiber . Therefore all fibers are irreducible, reduced, and of degree one in the corresponding embedding. It follows that for every fiber Therefore is a fiber bundle: , where is a rank vector bundle. Since is toric, this bundle splits as a sum of line bundles , and therefore is a strict Cayley polytope. Hence (1) implies (2).
Assume (2) holds. By Proposition 3.9 (with ), is -normal, and . Since is an integer, , hence , so . Therefore (1) holds.
The equivalence of (2) and (3) is essentially contained in ; here is a brief sketch of the proof.
Assume (2) holds. Since , is defective with defect [11, Prop. 5.12, p. 369] hence , since .
Assume (3) holds, so that is defective with defect . By [7, Thm. 2], then