On $G$-birational rigidity of del Pezzo surfaces

TL;DR

This paper proves that for smooth del Pezzo surfaces, $H$-birational rigidity implies $G$-birational rigidity for finite groups, confirming a geometric version of Kollár's question in dimension 2.

Contribution

It establishes that $H$-birational rigidity of del Pezzo surfaces implies $G$-birational rigidity for finite groups, extending previous understanding in algebraic geometry.

Findings

$H$-birational rigidity implies $G$-birational rigidity for del Pezzo surfaces.

Answers Kollár's question positively in dimension 2.

Provides a new perspective on group actions on algebraic surfaces.

Abstract

Let $G$ be a finite group and $H\subseteq G$ be its subgroup. We prove that if a smooth del Pezzo surface over an algebraically closed field is $H$-birationally rigid then it is also $G$-birationally rigid, answering a geometric version of Koll\'{a}r's question in dimension 2 by positive.