On terminal Fano 3-folds with 2-torus action

TL;DR

This paper classifies terminal Fano threefolds with a two-torus action, extending the toric Fano correspondence to broader classes using the anti canonical complex, which helps determine singularity types.

Contribution

It introduces the anti canonical complex for Fano varieties with complete intersection Cox rings, generalizing the toric correspondence and providing tools to analyze singularities.

Findings

Classification of terminal Fano threefolds with 2-torus action.

The anti canonical complex characterizes terminality and canonicity.

Application of tropical geometry to control discrepancies.

Abstract

We classify the terminal Fano threefolds of Picard number one that come with an effective action of a two-torus. Our approach applies also to higher dimensions and generalizes the correspondence between toric Fano varieties and lattice polytopes: to any Fano variety with a complete intersection Cox ring we associate its "anti canonical complex", which is a certain polyhedral complex living in the lattice of one parameter groups of an ambient toric variety. For resolutions constructed via the tropical variety, the lattice points inside the anticanonical complex control the discrepancies. This leads, for example, to simple characterizations of terminality and canonicity.