Classification of Gorenstein Toric Del Pezzo Varieties in arbitrary dimension

TL;DR

This paper provides a complete classification of Gorenstein toric Del Pezzo varieties in any dimension, identifying exactly 37 such varieties up to isomorphism, and relates these to Minkowski sum decompositions and almost Del Pezzo manifolds.

Contribution

It offers the first comprehensive biregular classification of Gorenstein toric Del Pezzo varieties in arbitrary dimensions, extending previous classifications.

Findings

Exactly 37 Gorenstein toric Del Pezzo varieties exist in each dimension

Classification relates to Minkowski sum decompositions of reflexive polygons

Connects to deformation classification of almost Del Pezzo manifolds

Abstract

A $n$-dimensional Gorenstein toric Fano variety $X$ is called Del Pezzo variety if the anticanonical class $-K_X$ is a $(n-1)$-multiple of a Cartier divisor. Our purpose is to give a complete biregular classfication of Gorenstein toric Del Pezzo varieties in arbitrary dimension $n \geq 2$. We show that up to isomorphism there exist exactly 37 Gorenstein toric Del Pezzo varieties of dimension $n$ which are not cones over $(n-1)$-dimensional Gorenstein toric Del Pezzo varieties. Our results are closely related to the classification of all Minkowski sum decompositions of reflexive polygons due to Emiris and Tsigaridas and to the classification up to deformation of $n$-dimensional almost Del Pezzo manifolds obtained by Jahnke and Peternell.