$G$-birational rigidity of the projective plane

TL;DR

This paper investigates the $G$-birational rigidity of the projective plane for all finite subgroups of PGL(3,C), establishing conditions under which the plane is rigid with respect to group actions.

Contribution

It provides a complete classification of the $G$-birational rigidity of the projective plane for every finite subgroup of PGL(3,C).

Findings

Determines when the projective plane is $G$-birationally rigid for all finite subgroups.

Classifies the $G$-rigidity based on subgroup properties.

Establishes criteria for the existence of $G$-birational maps between $G$-minimal surfaces.

Abstract

Given a surface $S$ and a finite group $G$ of automorphisms of $S$, consider the birational maps $S\dashrightarrow S'$ that commute with the action of $G$. This leads to the notion of a $G$-minimal variety. A natural question arises: for a fixed group $G$, is there a birational $G$-map between two different $G$-minimal surfaces? If no such map exists, the surface is said to be $G$-birationally rigid. This paper determines the $G$-rigidity of the projective plane for every finite subgroup $G\subset\mbox{PGL}_3\left(\mathbb{C}\right)$.