Equivalence classes for smooth Fano polytopes

Equivalence classes for smooth Fano polytopes

Akihiro Higashitani Akihiro Higashitani, Department of Mathematics, Kyoto University, Japan Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan ahigashi@math.kyoto-u.ac.jp
Abstract.

Let be the set of smooth Fano -polytopes up to unimocular equivalence. In this paper, we consider the F-equivalence or I-equivalence classes for and introduce F-isolated or I-isolated smooth Fano -polytopes. First, we describe all of F-equivalence classes and I-equivalence classes for . We also give a complete characterization of F-equivalence classes (I-equivalence classes) for smooth Fano -polytopes with vertices and construct a family of I-isolated smooth Fano polytopes.

2010 Mathematics Subject Classification: Primary 14M25; Secondary 52B20.
      Keywords: smooth Fano polytope, toric Fano manifold, F-equivalent, I-equivalent, primitive collection, primitive relation.

1. Introduction

Let be a lattice -polytope, i.e., a convex polytope of dimension whose vertices belong to . Let denote the set of the vertices of . We recall some notions on lattice polytopes.

  • We say that is reflexive if contains the origin in its interior and its dual polytope is also a lattice polytope, where stands for the usual inner product on .

  • We say that is simplicial if each facet of contains exactly vertices.

  • We say that is a smooth Fano polytope if the origin is contained in its interior and the vertex set of each facet of forms a -basis for . In particular, every smooth Fano polytope is reflexive and simplicial.

  • Two lattice polytopes and are unimodularly equivalent if there is an affine map such that and .

As is well known, each smooth Fano -polytope (up to unimodular equivalence) one-to-one corresponds to a toric Fano -fold (up to isomorphism). Thus, knowing smooth Fano polytopes is equivalent to knowning toric Fano manifolds in some sense.

By many researchers, smooth Fano polytopes or toric Fano manifolds have been investigated. Especially, their classification for each dimension is one of the most interesting problem. The complete classification of smooth Fano polytopes or toric Fano manifolds was given by [1] and [13] in dimension 3, by [3] and [12] in dimension 4 and by [8] in dimension 5. Recently, an explicit algorithm classifying all smooth Fano polytopes has been constructed by M. Øbro in 2007 (see [9] and [10]). The following table shows the number of unimodular equivalence classes for smooth Fano -polytopes.

1 2 3 4 5 6 7
smooth Fano -polytopes 1 5 18 124 866 7622 72256
Table 1. The number of unimodular equivalence classes of smooth Fano -polytopes with

Let be the set of all unimodular equivalence classes for smooth Fano -polytopes. The main concern of this paper is some equivalence classes for with respet to the following two equivalence relations:

Definition 1.1 ([12, Definition 1.1, 6.1], [11, Definition 1.1]).

We say that two smooth Fano -polytopes and are F-equivalent if there exists a sequence , , of smooth Fano -polytopes satisfying the following three conditions:

  • and are unimodularly equivalent to and , respectively;

  • For each , we have with or with for some lattice point ;

  • If , then there exists a proper face of such that and the set of facets of containing is equal to

    In other words, is obtained by taking a stellar subdivision of with . If , then the similar condition holds.

Definition 1.2 ([11, Section 1]).

Two smooth Fano -polytopes and are called I-equivalent if there exists a sequence of smooth Fano -polytopes satisfying the conditions (a) and (b) in Definition 1.2, i.e., (c) is not necessarily satisfied.

Clearly, if two smooth Fano polytopes and are F-equivalent, then those are also I-equivalent.

Remark 1.3.

(a) These definitions are also available to complete nonsingular fans by exchanging some terminologies with the ones for fans. For example, we may exchange the set of the vertices of a polytope into the set of the primitive vectors of 1-dimensional cones in a fan.

(b) The condition (c) described in Definition 1.1 is interpreted as the condition that two corresponding toric Fano manifolds and are related with some equivariant blow-up or equivariant blow-down. On the other hand, I-equivalence does not necessarily correspond to an equilvariant blow-up (blow-down).

Let be the unit coordinate vectors of and let

Then is the smooth Fano polytope -polytope corresponding to the -dimensional projective space . Note that this is the unique smooth Fano -polytope with vertices (up to unimodular equivalence). For a given positive integer , let

Then it is well known that these are smooth Fano -polytopes. Note that and are I-equivalent and each of them is also I-equivalent to , while and are NOT F-equivalent when .

We say that a polytope is pseudo-symmetric if contains a facet such that is also a facet of . By Ewald [5], it is proved that every pseudo-symmetric smooth Fano polytope is unimodularly equivalent to

where for two reflexive polytopes and , denotes the free sum of and , i.e.,

Note that the free sum of smooth Fano polytopes corresponds to the direct product of toric Fano manifolds.


Sato [12] investigated the F-equivalence classes for as follows: For every (resp. ), is F-equivalent to (resp. ), namely, each of and has the unique F-equivalence class. Moreover, consists of three F-equivalence classes, one of which consists of 122 smooth Fano 4-polytopes being F-equivalent to . Each of the others consists of one smooth Fano 4-polytope, which are and , respectively.

Sato also conjectured that any smooth Fano -polytope is either F-equivalent to or pseudo-symmetric. ([12, Conjecture 1.3 and 6.3]). This is true when . However, Øbro [11] has given a counterexample of this conjecture. His counterexample is a smooth Fano 5-polytope with 8 vertices which is neither pseudo-symmetric nor I-equivalent to any other smooth Fano 5-polytope, i.e., for any , is never I-equivalent to .

In this paper, for the further investigations of equivalence classes for with respect to both of F-equivalence and I-equivalence, we investigate smooth Fano 5-polytopes and determine all of F-equivalence classes as well as I-equivalence classes for . Moreover, we introduce F-isolated or I-isolated smooth Fano polytope (See Section 4) and we characterize completely F-isolated (I-isolated) smooth Fano -polytopes with vertices. In addition, we construct a family of I-isolated smooth Fano -polytopes for each .

A brief organization of this paper is as follows. First, in Section 2, we recall some notion on smooth Fano polytopes, fix some notation and prepare some lemmas for the main results. Next, in Section 3, we describe all of F-equivalence classes as well as I-equivalence classes for . Moreover, in Section 4, we introduce F-isolated or I-isolated smooth Fano polytopes and we give a complete characterization of F-isolated (I-isolated) smooth Fano -polytopes with vertices (Theorem 4.2). In addition, we construct a family of I-isolated smooth Fano -polytopes with vertices for and (Theorem 4.3 and Corollary 4.6).

2. Preliminaries

First, we note the following. For a smooth Fano polytope , let

where denotes the cone generated by . Then is a complete nonsingular fan. (For the terminologies on fans, conslut, e.g., [6].)

Let be a complete nonsingular fan. We recall the useful notions, primitive collections and primitive relations, introduced by Batyrev [2]. Let be the set of the primitive vectors of 1-dimensional cones in .

  • We call a nonempty subset a primitive collection of if is not a cone in but is a cone in for every . Let PC denote the set of all primitive collections of .

  • Let . If , then there is a unique cone in such that where . We call the relation or the primitive relation for .

  • For , if the primitive relation for is , then is called the degree of .

Notice that these definitions are also available to smooth Fano polytopes. In the case of smooth Fano polytopes , we use a notation PC instead of PC.

Proposition 2.1 ([12, Theorem 3.10], [3]).

For a complete nonsingular fan , the degrees of the primitive collections of are all positive if and only if there exists a smooth Fano polytope such that .

Moreover, the primitive relations in smooth Fano -polytopes with or vertices are completely characterized as follows.

Proposition 2.2 ([7, Theorem 1]).

Let be a smooth Fano -polytope with vertices and let . Then the primitive relations in are (up to renumeration of the vertices) of the form

Proposition 2.3 ([2, Theorem 6.6]).

Let be a smooth Fano -polytope with vertices. Then one of the following holds:

  • consists of three disjoint primitive collections, i.e., with for each ;

  • and there is with such that the primitive relations in are of the forms

    (1)

    where and and the degree of each of these primitive collections is positive.

We also recall the following useful lemma.

Lemma 2.4 ([11, Lemma 3.1], see also [4, Corollary 4.4]).

Let

(2)

be a linear relation of some vertices of a smooth Fano polytope such that and . Suppose that and is a face of . Then (2) is a primitive relation. Moreover, for each face of with , is also a face of for every .

3. Equivalence classes for smooth Fano 5-polytopes

In this section, we describe all F-equivalence or I-equivalence classes for .

Proposition 3.1.

The number of F-equivalence classes for is and the number of I-equivalence classes for is .

The following Figure 1 shows all of the smooth Fano 5-polytopes which are not F-equivalent to . Each circled number corresponds to one smooth Fano 5-polytope with vertices for and the number is the ID of the database “Graded Ring Database” of smooth Fano polytopes, which is based on the algorithm by Øbro ([9, 10]). See http://grdb.lboro.ac.uk/forms/toricsmooth

There are 2 smooth Fano 5-polytopes with 8 vertices, 12 ones with 9 vertices, 16 ones with 10 vertices, 6 ones with 11 vertices and 2 ones with 12 vertices which are not F-equivalent to . See also the table 2. Moreover, each of the double circled numbers corresponds to a smooth Fano 5-polytope such that for any , is not I-equivalent to (i.e., I-isolated, see Section 4). Note that Øbro’s example [11] corresponds to the double circled number 164. In addition, if two circled numbers are connected by a line, then those are F-equivalent. We can see that there are 26 “connected components” in Figure 1, each of which corresponds to an F-equivalence class for . On the other hand, all of the smooth Fano 5-polytopes corresponding to the single circled numbers are I-equivalent to .

Although there are only 2 smooth Fano 4-polytopes which are not F-equivalent to , which are and (see [12]), there are 38 smooth Fano 5-polytopes which are not F-equivalent to , i.e., there are 38 circled numbers in Figure 1.


Figure 1. smooth Fano 5-polytopes which are not F-equivalent to
3 4 5 6 7
smooth Fano -polytopes 2 12 16 6 2
Table 2. The number of smooth Fano 5-polytopes with vertices which are not F-equivalent to

4. I-isolated smooth Fano polytopes

We introduce the notions, F-isolated and I-isolated, for smooth Fano polytopes.

Definition 4.1.

Let be a smooth Fano -polytope. We say that is F-isolated (resp. I-isolated) if for any , is not F-equivalent (resp. I-equivalent) to .

Obviously, is F-isolated if is I-isolated.

First, we characterize the primitive relations in I-isolated smooth Fano -polytopes with vertices. Note that such a polytope is of dimension at least 5.

Theorem 4.2.

Let be a smooth Fano -polytope with vertices. Then the following three conditions are equivalent:

  • is F-isolated;

  • is I-isolated;

  • All of the primitive relations in are of the forms

    (3)

    where , and with .

Proof.

((a) (c)) Assume that is F-isolated. Since , from [12, Corllary 6.13], we may assume satisfies the conditions in Proposition 2.3 (ii), i.e., the primitive relations in are of the forms (1).

In the discussions below, by using the description of the primitive relations in a complete nonsingular fan obtained by introducing a new lattice point for some and taking a stellar subdivision with the 1-dimensional cone generated by , we prove that if there is no smooth Fano polytope which is F-equivalent to , then the primitive relations in is of the forms (3). The description is explicitly given in [12, Theorem 4.3].

For a face , let denote the new complete nonsingular fan obtained by taking a stellar subdivision of with , where .

  • Suppose or . Then by [12, Proposition 8.3], we obtain a new smooth Fano -polytope with vertices which is F-equivalent to , a contradiction. Hence and .

  • Suppose . Since is not contained in PC, is a face of . We consider a stellar subdivision of with a lattice point and the complete nonsingular fan . Then the new primitive relations in concerning are

    (See [12, Theorem 4.3].) Since the degree of each of these new primitive collections is positive, by Proposition 2.1, there is a smooth Fano polytope with , and in particular, is F-equivalent to , a contradiction. Hence .

  • Suppose . Then we see that the complete nonsingular fan comes from a smooth Fano polytope, a contradiction. Thus . On the other hand, since is a primitive collection of and its degree is equal to , we obtain that by Propoisition 2.1. Hence .

  • Similarly, suppose . Then we see that the complete nonsingular fan (P) comes from a smooth Fano polytope, a contradiction. Thus . Moreover, since is a primitive collection of and its degree is equal to , we have by Proposition 2.1. Hence .

  • Hence, in particular, we obtain . This implies that and .

  • Suppose . Then we see that the complete nonsingular fan comes from a smooth Fano polytope, a contradiction. Thus . On the other hand, since is a primitive collection of and its degree is equal to , we obtain that by Proposition 2.1. Hence .

By summarizing these, we obtain the desired primitive relations (3).

(c) (b) Let

let and let . Then the primitive relations in are of the forms (3). Our work is to prove that this is I-isolated.

: First, we prove there is no smooth Fano -polytope with vertices such that .

Suppose, on the contrary, that there is a smooth Fano -polytope with vertices such that , where .

Assume that . In this case, although there must be a primitive collection PC with by Proposition 2.2, no non-empty subset of add to 0, a contradiction.

Assume that . Then for some . On the other hand, for each (resp. ), the th entry of each vertex of except for is nonpositive (resp. nonnegative). Thus cannot contain the origin in its interior, a contradiction.

Assume that . Let be a primitive collection of with . Then . By Proposition 2.2, should be written as a linear combination of and , a contradiction.

: Next, we prove there is no smooth Fano -polytope with vertices such that .

Suppose that there is a smooth Fano -polytope with vertices such that for some new lattice point .

Since is a vertex of , by Lemma 2.4, the relation

(4)

is a primitive relation in . Hence is a face of for every . Since we have the relation

(5)

by Lemma 2.4, we obtain that (5) is also a primitive relation in and

are the facets of for every and . Moreover, since is a face of , we obtain that

is a facet of for every and by (4). In addition, from the relation

(6)

since is a face of , we obtain that (6) is also a primitive relation in and

is also a facet of for every and .

Therefore, contains the following four kinds of facets:

These are also the facets of . Thus, for any these facets , is not contained in . Therefore, is contained in the cone generated by the remaining facet of , i.e.,

for some and . Without loss of generality, we may assume that . Let

where . Let , where and . Then is a facet of . Let be the unique facet of such that is a ridge (i.e. the face of dimension ) of with . Then it must be satisfied that . In fact, for each , cannot be a face of because and cannot be a face, and for , is not a face by our assumption. Hence, by [11, Lemma 2.1], is in the linear subspace spanned by . Therefore, from the relation , we have . Moreover it follows from [11, Lemma 2.1] again that we have

where is the lattice vector defining , i.e., . Thus . Namely, can be written like with some . Let . Since is a facet of for each and the relation holds, we see that is also contained in , i.e., is not simplicial, a contradiction.

((b) (a)) This is obvious. ∎

Next, we provide a family of I-isolated smooth Fano polytopes.

Theorem 4.3.

Let , , , and for be integers. Then there exists an I-isolated (in particular, F-isolated) smooth Fano polytope of dimension with vertices whose primitive relations are of the forms

(7)

where and .

Proof.

Let

let and let for each and . We define . Then is a smooth Fano polytope of dimension with vertices and all of its primitive relations are of the forms (7). By Lemma 4.4 and Lemma 4.5 below, we see that is I-isolated. ∎

Lemma 4.4.

Let be the polytope given in the proof of Theorem 4.3. Then there exists no smooth Fano -polytope with vertices such that .

Proof.

Work with the same notation as in the proof of Theorem 4.3. Suppose that there is a smooth Fano -polytope with vertices such that , where .

Assume that . In this case, although there must be a primitive collection PC with by Proposition 2.2, no non-empty subset of add to 0, a contradiction.

Assume that . Then for some . On the other hand, for each (resp. ), the th entry of each vertex of except for is nonpositive (resp. nonnegative). Thus cannot contain the origin in its interior, a contradiction. Similarly, if , then cannot contain the origin in its interior, a contradiction.

Assume that . Without loss of generality, we assume . For each , let and let . Let be an index attaining . Consider . Then is a face of . In fact, let

Then we see the following:

  • We have for each , , for each and for each and .

  • Since , we have

  • We have