Automorphisms of singular three-dimensional cubic hypersurfaces

TL;DR

This paper classifies certain singular cubic threefolds with finite group actions, excluding rational and birationally equivalent cases, and proves the Segre cubic's $A_5$-birational superrigidity.

Contribution

It provides a classification of singular cubic threefolds with group actions under specific non-rationality and birationality conditions, and establishes the $A_5$-birational superrigidity of the Segre cubic.

Findings

Classification of non-$G$-rational singular cubic threefolds with group actions.

Proof of $A_5$-birational superrigidity of the Segre cubic.

Exclusion of certain birational structures for these hypersurfaces.

Abstract

In this paper we classify three-dimensional singular cubic hypersurfaces with an action of a finite group $G$, which are not $G$-rational, are not $G$-birationally isomorphic to a quadric and have no birational structure of $G$-Mori fiber space with the base of positive dimension. Also we prove the $A_{5}$ -birational superrigidity of the Segre cubic.