Rationality of the quotient of $\mathbb{P}^2$ by finite group of automorphisms over arbitrary field of characteristic zero

TL;DR

This paper proves that the quotient of the projective plane by a finite automorphism group over any field of characteristic zero is rational, extending known results from algebraically closed fields to arbitrary fields.

Contribution

It establishes the rationality of the quotient of the projective plane by finite automorphism groups over any characteristic zero field, generalizing previous algebraically closed field results.

Findings

The quotient $P^2_K / G$ is rational over any characteristic zero field.

Extends Castelnuovo's rationality criterion to arbitrary fields.

Provides new insights into automorphism group actions on projective planes.

Abstract

Let $\Bbbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\Bbbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\Bbbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\Bbbk} / G$ is rational for an arbitrary field $\Bbbk$ of characteristic zero.