Multigraded Factorial Rings and Fano varieties with torus action

TL;DR

This paper classifies and constructs Fano varieties with torus actions using multigraded factorial rings, providing explicit bounds, lists, and Cox rings for various dimensions and Picard indices.

Contribution

It describes all finitely generated factorial algebras with multigrading of complexity one, enabling systematic construction and classification of related Fano varieties.

Findings

15 non-toric surfaces with Picard index ≤6

116 non-toric threefolds with Picard index ≤2

67 non-toric locally factorial fourfolds

Abstract

In a first result, we describe all finitely generated factorial algebras over an algebraically closed field of characteristic zero that come with an effective multigrading of complexity one by means of generators and relations. This enables us to construct systematically varieties with free divisor class group and a complexity one torus action via their Cox rings. For the Fano varieties of this type that have a free divisor class group of rank one, we provide explicit bounds for the number of possible deformation types depending on the dimension and the index of the Picard group in the divisor class group. As a consequence, one can produce classification lists for fixed dimension and Picard index. We carry this out expemplarily in the following cases. There are 15 non-toric surfaces with Picard index at most six. Moreover, there are 116 non-toric threefolds with Picard index at most two; nine of them are locally factorial, i.e. of Picard index one, and among these one is smooth, six have canonical singularities and two have non-canonical singularities. Finally, there are 67 non-toric locally factorial fourfolds and two one-dimensional families of non-toric locally factorial fourfolds. In all cases, we list the Cox rings explicitly.