Quotients of del Pezzo surfaces of degree 2

TL;DR

This paper investigates the rationality of quotients of degree 2 del Pezzo surfaces under group actions over fields of characteristic zero, identifying specific groups that produce non-rational quotients and providing explicit examples.

Contribution

It classifies groups acting on degree 2 del Pezzo surfaces that yield non-$bbk$-rational quotients and constructs explicit rational and non-rational examples for these cases.

Findings

Certain groups produce non-$bbk$-rational quotients.

For most groups, the quotient surface remains $bbk$-rational.

Explicit examples of rational and non-rational quotients are provided.

Abstract

Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface of degree~$2$ and $G$ be a group acting on $X$. In this paper we study $\Bbbk$-rationality questions for the quotient surface $X / G$. If there are no smooth $\Bbbk$-points on $X / G$ then $X / G$ is obviously non-$\Bbbk$-rational. Assume that the set of smooth $\Bbbk$-points on the quotient is not empty. We find a list of groups, such that the quotient surface can be non-$\Bbbk$-rational. For these groups we construct examples of both $\Bbbk$-rational and non-$\Bbbk$-rational quotients of both $\Bbbk$-rational and non-$\Bbbk$-rational del Pezzo surfaces of degree $2$ such that the $G$-invariant Picard number of $X$ is $1$. For all other groups we show that the quotient $X / G$ is always $\Bbbk$-rational.