Quotients of cubic surfaces

TL;DR

This paper investigates the rationality of quotient surfaces obtained from cubic surfaces under group actions, establishing conditions for rationality and providing explicit examples for the case of groups of order three.

Contribution

It proves that quotients of cubic surfaces by nontrivial groups (except certain order 3 cases) are rational, and constructs examples illustrating the complexity for order 3 groups.

Findings

Quotients are rational when group order is not 3 under certain conditions.

Explicit examples of rational and nonrational quotients for order 3 groups.

Conditions for rationality depend on the group action and surface properties.

Abstract

Let $\Bbbk$ be any field of characteristic zero, $X$ be a cubic surface in $\mathbb{P}^3_{\Bbbk}$ and $G$ be a group acting on $X$. We show that if $X(\Bbbk) \ne \varnothing$ and $G$ is not trivial and not a group of order $3$ acting in a special way then the quotient surface $X / G$ is rational over $\Bbbk$. For the group $G$ of order $3$ we construct examples of both rational and nonrational quotients of both rational and nonrational $G$-minimal cubic surfaces over $\Bbbk$.