Tohoku Mathematical Journal

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

Benjamin Nill and Mikkel Øbro

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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.

Article information

Tohoku Math. J. (2), Volume 62, Number 1 (2010), 1-15.

First available in Project Euclid: 31 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 14J45: Fano varieties 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]

toric varieties Fano varieties lattice polytopes


Nill, Benjamin; Øbro, Mikkel. $\boldsymbol{Q}$-factorial Gorenstein toric Fano varieties with large Picard number. Tohoku Math. J. (2) 62 (2010), no. 1, 1--15. doi:10.2748/tmj/1270041023.

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