Canonical divisors on T-varieties

TL;DR

This paper extends toric geometry to study compact varieties with lower-dimensional torus actions, describing divisors via convex geometry and classifying certain Fano varieties with small torus actions.

Contribution

It introduces a convex geometric framework for divisors on T-varieties and classifies specific log del Pezzo surfaces, advancing the understanding of Fano varieties with torus actions.

Findings

Classification of log del Pezzo C*-surfaces of Picard number 1

Explicit descriptions of equivariant smoothings of Fano threefolds

Criteria for ampleness of divisors on T-varieties

Abstract

Generalising toric geometry we study compact varieties admitting lower dimensional torus actions. In particular we describe divisors on them in terms of convex geometry and give a criterion for their ampleness. These results may be used to study Fano varieties with small torus actions. As a first result we classify log del Pezzo C*-surfaces of Picard number 1 and Gorenstein index less than 4. In further examples we show how classification might work in higher dimensions and we give explicit descriptions of some equivariant smoothings of Fano threefolds.