G-birationally rigid cubic threefolds

TL;DR

This paper classifies cubic threefolds with finite automorphism groups that are birationally rigid under the group action, identifying when such rigidity occurs.

Contribution

It provides a classification of pairs (X,G) where X is a cubic threefold and G a finite subgroup, establishing conditions for G-birational rigidity.

Findings

Identifies all pairs (X,G) with G-birationally rigid cubic threefolds.

Establishes criteria for G-birational rigidity in cubic threefolds.

Shows that certain automorphism groups enforce rigidity.

Abstract

We classify pairs $(X,G)$ consisting of a (possibly singular) cubic threefold $X\subset\mathbb{P}^4$ and a finite subgroup $G\subset\mathrm{Aut}(X)$ such that $X$ is $G$-birationally rigid, i.e., $X$ is a $G$-Mori fiber space (over a point), and $X$ is not $G$-birational to any $G$-Mori fibre space that is not $G$-biregular to $X$.