The number of vertices of a Fano polytope

The Pseudo-Index of Horospherical Fano Varieties

“…Moreover, they have proved Theorem 1 in the case of Fano varieties of dimension 3 and 4, all flag varieties, and also for some particular toric varieties. More generally, this result has been also proved by C. Casagrande for all Q-factorial toric Fano varieties [5]. Theorem 1 generalizes these results to the family of 1…”

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

“…For every d ≥ 1 there are finitely many isomorphism classes of reflexive d-polytopes, and for d ≤ 4 they have been completely classified using computer algorithms ( [10], [11]). Simplicial reflexive d-polytopes have at most 3d vertices ( [6] theorem 1). This upper bound is attained if and only if d is even and P splits into d 2 copies of del Pezzo 2-polytopes V 2 = conv{±e 1 , ±e 2 , ±(e 1 − e 2 )}, where {e 1 , e 2 } is a basis of a 2-dimensional lattice.…”

Fano polytopes

“…Their combinatorics is quite restrictive: The vertex-edge-graph has diameter two, and given any vertex v 2 V(P ) there exist at most three other vertices of P not contained in a facet containing v [40]. Casagrande [14] showed that the maximal number of vertices is 3n if n is even, or 3n 1 if n is odd. Equivalently, this bounds the rank of the Picard group of the corresponding variety X, since rk Pic X = |V(P )| dim P .…”

“…This initiated an intense study of the geometric and combinatorial properties of reflexive polytopes [14,19,31,32,40,42]. Definition 2.1.…”