The automorphism group of a variety with torus action of complexity one

TL;DR

This paper investigates the automorphism groups of certain algebraic varieties with torus actions, explicitly describing their root systems and applying findings to classify specific almost homogeneous Fano varieties.

Contribution

It provides an explicit description of the roots of automorphism groups for varieties with torus actions of complexity one, advancing understanding of their structure and classification.

Findings

Determined roots of automorphism groups for these varieties.

Explicitly described the root system of the semisimple part.

Classified almost homogeneous del Pezzo surfaces and Fano threefolds with specific properties.

Abstract

We consider a normal complete rational variety with a torus action of complexity one. In the main results, we determine the roots of the automorphism group and give an explicit description of the root system of its semisimple part. The results are applied to the study of almost homogeneous varieties. For example, we describe all almost homogeneous (possibly singular) del Pezzo k*-surfaces of Picard number one and all almost homogeneous (possibly singular) Fano threefolds of Picard number one having a reductive automorphism group with two-dimensional maximal torus.