Translator Disclaimer
2015 On smooth Gorenstein polytopes
Benjamin Lorenz, Benjamin Nill
Tohoku Math. J. (2) 67(4): 513-530 (2015). DOI: 10.2748/tmj/1450798070


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.


Download Citation

Benjamin Lorenz. Benjamin Nill. "On smooth Gorenstein polytopes." Tohoku Math. J. (2) 67 (4) 513 - 530, 2015.


Published: 2015
First available in Project Euclid: 22 December 2015

zbMATH: 1338.52014
MathSciNet: MR3436539
Digital Object Identifier: 10.2748/tmj/1450798070

Primary: 52B20
Secondary: 14J45, 14M25

Rights: Copyright © 2015 Tohoku University


Vol.67 • No. 4 • 2015
Back to Top