Tohoku Mathematical Journal

On smooth Gorenstein polytopes

Benjamin Lorenz and Benjamin Nill

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A Gorenstein polytope of index $r$ is a lattice polytope whose $r$th dilate is a reflexive polytope. These objects are of interest in combinatorial commutative algebra and enumerative combinatorics, and play a crucial role in Batyrev's and Borisov's computation of Hodge numbers of mirror-symmetric generic Calabi-Yau complete intersections. In this paper we report on what is known about smooth Gorenstein polytopes, i.e., Gorenstein polytopes whose normal fan is unimodular. We classify $d$-dimensional smooth Gorenstein polytopes with index larger than $(d+3)/3$. Moreover, we use a modification of Øbro's algorithm to achieve classification results for smooth Gorenstein polytopes in low dimensions. The first application of these results is a database of all toric Fano $d$-folds whose anticanonical divisor is divisible by an integer $r$ satisfying $r \ge d-7$. As a second application we verify that there are only finitely many families of Calabi-Yau complete intersections of fixed dimension that are associated to a smooth Gorenstein polytope via the Batyrev-Borisov construction.

Article information

Tohoku Math. J. (2), Volume 67, Number 4 (2015), 513-530.

First available in Project Euclid: 22 December 2015

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Zentralblatt MATH identifier

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

Gorenstein polytopes smooth reflexive polytopes toric varieties Fano manifolds Calabi-Yau manifolds


Lorenz, Benjamin; Nill, Benjamin. On smooth Gorenstein polytopes. Tohoku Math. J. (2) 67 (2015), no. 4, 513--530. doi:10.2748/tmj/1450798070.

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