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Tohoku Mathematical Journal
2010
March
SECOND SERIES VOL. 62, NO. 1
Tohoku Math. J.
62 (2010), 115

Title
$\boldsymbol{Q}$FACTORIAL GORENSTEIN TORIC FANO VARIETIES WITH LARGE PICARD NUMBER
Author
Benjamin Nill and Mikkel Øbro
(Received January 23, 2009, revised September 2, 2009) 
Abstract.
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 $3d1$ 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 $3d1$ vertices, corresponding to $d$dimensional ${\boldsymbol Q}$factorial Gorenstein toric Fano varieties with Picard number $2d1$. 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.
2000 Mathematics Subject Classification.
Primary 14M25; Secondary 14J45, 52B20.


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