A bound for the splitting of smooth Fano polytopes with many vertices

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.