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Q-factorial Gorenstein toric Fano varieties with large Picard number

Q-factorial Gorenstein toric Fano varieties X of dimension d with Picard number rho(X) correspond to simplicial reflexive d-polytopes with rho(X)+d vertices. Casagrande showed that any simplicial reflexive d-polytope has at most 3d vertices, if d is even, respectively, 3d-1, if d is odd. Moreover, it is known that equality for d even implies uniqueness up to unimodular equivalence. In this paper we completely classify all simplicial reflexive d-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 odd (d > 1) there exist precisely two, both being nonsingular toric fiber bundles over the projective line. This generalizes recent work of the second author.