Q-factorial Gorenstein Toric Fano Varieties with Large Picard Number

In dimension d, Q-factorial Gorenstein toric Fano varieties with Picard number ρX correspond to simplicial reflexive polytopes with ρX+d vertices. Casagrande showed that any d-dimensional simplicial reflexive polytope has at most 3d vertices, if d is even, respectively, 3d − 1, if d is odd. Moreover, for d even there is up to unimodular equivalence only one such polytope with 3d vertices, corresponding to (S3) with Picard number 2d, where S3 is the blow-up of P at three non collinear points. In this paper we completely classify all d-dimensional simplicial reflexive polytopes having 3d − 1 vertices, corresponding to d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number 2d− 1. For d even, there exist three such varieties, with two being singular, while for d > 1 odd there exist precisely two, both being nonsingular toric fiber bundles over P. This generalizes recent work of the second author.