2014/09/25 by Assarf, Benjamin, Nill, Benjamin
#14J45 #14M25 #52B20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1409.7303
The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior such that the vertex set of each facet forms a lattice basis. Casagrande showed that any smooth d-dimensional Fano polytope has at most 3d vertices. Smooth Fano polytopes in dimension d with at least 3d-2 vertices are completely known. The main result of this paper deals with the case of 3d-k vertices for k fixed and d large. It implies that there is only a finite number of isomorphism classes of toric Fano d-folds X (for arbitrary d) with Picard number 2d-k such that X is not a product of a lower-dimensional toric Fano manifold and the projective plane blown up in three torus-invariant points. This verifies the qualitative part of a conjecture in a recent paper by the first author, Joswig, and Paffenholz.