On a combinatorial description of the Gorenstein index for varieties with torus action

TL;DR

This paper demonstrates that the Gorenstein index of certain Fano varieties with torus action can be determined from their anticanonical complex using lattice distances, extending toric geometry concepts.

Contribution

It introduces a combinatorial method to compute the Gorenstein index for varieties with torus action via the anticanonical complex, generalizing toric Fano polytope techniques.

Findings

Gorenstein index can be read from the anticanonical complex

Provides bounds on data of Fano threefolds with specific automorphism groups

Extends combinatorial tools from toric to more general varieties

Abstract

The anticanonical complex is a combinatorial tool that was invented to extend the features of the Fano polytope from toric geometry to wider classes of varieties. In this note we show that the Gorenstein index of Fano varieties with torus action of complexity one (and even more general of the so-called general arrangement varieties) can be read off its anticanonical complex in terms of lattice distances in full analogy to the toric Fano polytope. As an application we give concrete bounds on the defining data of almost homogeneous Fano threefolds of Picard number one having a reductive automorphism group with two-dimensional maximal torus depending on their Gorenstein index.