The Classification of Fano 3-Folds with Torus Embeddings

Abstract

Let X be a complex smooth Fano variety of dimension at least four. In this paper, we classify such X when the pseudoindex is at least n − 2 and the Picard number greater than one. We also discuss the relations between pseudoindex and other invariants of Fano varieties.

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.…”

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.…”