# Quotients of del Pezzo surfaces

## TL;DR

This paper investigates when quotients of degree 1 del Pezzo surfaces by finite groups are non-rational over a field of characteristic zero, providing classifications and examples for specific group actions.

## Contribution

It classifies non-$bbk$-rational quotients of degree 1 del Pezzo surfaces under certain group actions, extending previous work on higher degrees.

## Key findings

- Non-$bbk$-rational quotients occur only for trivial, cyclic groups of order 2, 3, and 6.
- Both $X$ and $X/G$ are non-$bbk$-rational if the $G$-invariant Picard number is 1 for trivial and order 2 groups.
- Examples show that quotients can be $bbk$-rational or not, regardless of the rationality of the original surface.

## Abstract

Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface and $G$ be a finite subgroup in $\operatorname{Aut}(X)$. In this paper we study when the quotient surface $X / G$ can be non-rational over $\Bbbk$. Obviously, if there are no smooth $\Bbbk$-points on $X / G$ then it is not $\Bbbk$-rational. Therefore under assumption that the set of smooth $\Bbbk$-points on $X / G$ is not empty we show that there are few possibilities for non-$\Bbbk$-rational quotients. The quotients of del Pezzo surfaces of degree $2$ and greater are considered in the author's previous papers. In this paper we study the quotients of del Pezzo surfaces of degree $1$. We show that they can be non-$\Bbbk$-rational only for the trivial group or cyclic groups of order $2$, $3$ and $6$. For the trivial group and the group of order $2$ we show that both $X$ and $X / G$ are not $\Bbbk$-rational if the $G$-invariant Picard number of $X$ is $1$. For the groups of order $3$ and $6$ 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 $1$ such that the $G$-invariant Picard number of $X$ is $1$. As a result of complete classification of non-$\Bbbk$-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of $\Bbbk$-rational surfaces, and obtain some corollaries concerning fields of invariants of $\Bbbk(x , y)$.