Tohoku Mathematical Journal
2015
December
SECOND SERIES VOL. 67, NO. 4

Tohoku Math. J.
67 (2015), 513-530

Title ON SMOOTH GORENSTEIN POLYTOPES

Author Benjamin Lorenz and Benjamin Nill

(Received February 26, 2014, revised July 17, 2014)
Abstract. 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.

Mathematics Subject Classification. Primary 52B20; Secondary 14M25, 14J45.
Key words and phrases. Gorenstein polytopes, smooth reflexive polytopes, toric varieties, Fano manifolds, Calabi-Yau manifolds.

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