Fano threefolds with 2-torus action - a picture book

TL;DR

This paper provides a combinatorial framework for classifying smooth Fano threefolds with 2-torus actions, enabling the computation of invariants and analysis of geometric properties.

Contribution

It introduces a new combinatorial description for these Fano threefolds, facilitating the study of their properties and invariants.

Findings

Identified conditions for the existence of Kahler-Einstein metrics.

Computed Cox rings for specific Fano threefolds.

Analyzed toric canonical degenerations of these varieties.

Abstract

Following the work of Altmann and Hausen we give a combinatorial description in terms for smooth Fano threefolds admitting a 2-torus action. We show that a whole variety of properties and invariants can be read off from this description. As an application we prove and disprove the existence of Kahler-Einstein metrics for some of these Fano threefolds, calculate their Cox rings and some of their toric canonical degenerations.