Fixed points of a finite subgroup of the plane Cremona group

TL;DR

This paper classifies all finite subgroups of the plane Cremona group that have a fixed point, identifying the associated rational surfaces with group actions fixing a point, thereby advancing understanding of group actions on rational surfaces.

Contribution

It provides a complete classification of finite subgroups with fixed points in the plane Cremona group, a significant step in understanding group actions on rational surfaces.

Findings

Complete classification of finite subgroups with fixed points

Identification of rational surfaces with fixed point group actions

Characterization of equivariant birational models

Abstract

We classify all finite subgroups of the plane Cremona group which have a fixed point. In other words, we determine all rational surfaces X with an action of a finite group G such that X is equivariantly birational to a surface which has a G-fixed point.