$\boldsymbol{Q}$-factorial Gorenstein toric Fano varieties with large Picard number

概要

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In dimension $d$, ${\boldsymbol Q}$-factorial Gorenstein toric Fano varieties with Picard number $\rho_X$ correspond to simplicial reflexive polytopes with $\rho_X + d$ vertices. Casagrande showed that any $d$-dimensional simplicial reflexive polytope has at most $3 d$ and $3d-1$ vertices if $d$ is even and odd, respectively. Moreover, for $d$ even there is up to unimodular equivalence only one such polytope with $3 d$ vertices, corresponding to the product of $d/2$ copies of a del Pezzo surface of degree six. In this paper we completely classify all $d$-dimensional simplicial reflexive polytopes having $3d-1$ vertices, corresponding to $d$-dimensional ${\boldsymbol Q}$-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 the projective line. This generalizes recent work of the second author.