On smooth Gorenstein polytopes

TL;DR

This paper classifies smooth Gorenstein polytopes, explores their properties, and applies these results to classify related toric Fano varieties and Calabi-Yau complete intersections, advancing understanding in algebraic geometry and combinatorics.

Contribution

It provides a classification of smooth Gorenstein polytopes with large index and develops algorithms for low-dimensional cases, linking these polytopes to toric Fano varieties and Calabi-Yau intersections.

Findings

Classified d-dimensional smooth Gorenstein polytopes with index > (d+3)/3.

Created a database of toric Fano d-folds with divisible anticanonical divisors.

Proved finiteness of Calabi-Yau families associated to these polytopes.

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 Oebro'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 larger than d-8. 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.