Tohoku Mathematical Journal

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

Benjamin Nill and Mikkel Øbro

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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 $3d-1$ 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

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

Dates
First available in Project Euclid: 31 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1270041023

Digital Object Identifier
doi:10.2748/tmj/1270041023

Mathematical Reviews number (MathSciNet)
MR2654299

Zentralblatt MATH identifier
1211.14048

Subjects
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]

Keywords
toric varieties Fano varieties lattice polytopes

Citation

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. https://projecteuclid.org/euclid.tmj/1270041023


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Tohoku University, Mathematical Institute

Tohoku Mathematical Journal

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