Toward the classification of higher-dimensional toric Fano varieties

Abstract

The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blow-ups and blow-downs, and get an easy criterion to determine whether a given nonsingular toric variety is a Fano variety or not. As applications of these results, we get a toric version of a theorem of Mori, and can classify, in principle, all nonsingular toric Fano varieties obtained from a given nonsingular toric Fano variety by finite successions of equivariant blow-ups and blow-downs through nonsingular toric Fano varieties. Especially, we get a new method for the classification of nonsingular toric Fano varieties of dimension at most four. These methods are extended to the case of Gorenstein toric Fano varieties endowed with natural resolutions of singularities. Especially, we easily get a new method for the classification of Gorenstein toric Fano surfaces. Proposition 2.2 (Batyrev [3])If ∆ ⊂ N R is a reflexive polytope, then T N emb(Σ(∆)) is a Gorenstein toric Fano variety. Conversely, if Σ is a finite complete fan in N such that T N emb(Σ) is a Gorenstein toric Fano variety, then Conv( G (Σ)) ⊂ N R is a reflexive polytope, where Conv( G (Σ)) is the convex hull of G (Σ) ⊂ N R . Moreover, any two Gorenstein toric Fano varieties T N emb(Σ(∆ 1 )) and T N emb(Σ(∆ 2 )) corresponding to two reflexive polytopes ∆ 1 ⊂ N R and ∆ 2 ⊂ N R are isomorphic if and only if ∆ 1 and ∆ 2 are equivalent up to unimodular transformation of the lattice N.Remark 2.3 A reflexive polytope ∆ is called a Fano polytope if Σ(∆) is nonsingular. Primitive collections and primitive relationsPrimitive collections and primitive relations, introduced by Batyrev [4], are very convenient in describing higher-dimensional fans. So in this section, we recall these concepts and characterize toric Fano varieties using them.

The number of vertices of a Fano polytope

“…For X smooth and toric, (ii) was already known in the cases n 7 or ι X 1 3 n + 1 [5]. For a smooth toric Fano X, (i) was conjectured by V. V. Batyrev (see [10, page 337]) and was already known to hold up to dimension 5 (for n 4 thanks to the classifications [2,17,4,15], and for n = 5 it is [6, Theorem 4.2]). Recently B. Nill [13] has extended this conjecture to the Q-factorial Gorenstein case, and has shown (i) for a certain class of Qfactorial, Gorenstein toric Fano varieties (see on page 124).…”

Gorenstein toric Fano varieties

“…Using a computer program Kreuzer There are many papers devoted to the investigation and classification of nonsingular toric Fano varieties [WW82,Bat82,Bat99,Sat00,Deb01,Cas03a,Cas03b]. In this article we present new classification results, bounds of invariants and conjectures concerning Gorenstein toric Fano varieties by investigating combinatorial and geometrical properties of reflexive polytopes.…”

“…In sections 2 and 3 two elementary technical tools are investigated and generalised that were previously already successfully used to investigate and classify nonsingular toric Fano varieties [Bat99,Sat00,Cas03b].…”

Classification of terminal simplicial reflexive d-polytopes with 3d − 1 vertices

“…An important subclass of terminal simplicial reflexive polytopes is the class of smooth reflexive polytopes, also known as smooth Fano polytopes: A reflexive polytope P is called smooth if the vertices of every face F of P is a part of a basis of the lattice N . Smooth Fano d-polytopes have been intensively studied and completely classified up to dimension 4 ([1], [4], [14], [16]). In higher dimensions not much is known.…”