Quotients of del Pezzo surfaces of high degree

Abstract

In this paper we study quotients of del Pezzo surfaces of degree four and more over arbitrary field $\mathbb{k}$ of characteristic zero by finite groups of automorphisms. We show that if a del Pezzo surface $X$ contains a point defined over the ground field and the degree of $X$ is at least five then the quotient is always $\mathbb{k}$-rational. If the degree of $X$ is equal to four then the quotient can be non-$\mathbb{k}$-rational only if the order of the group is $1$, $2$ or $4$. For these groups we construct examples of non-$\mathbb{k}$-rational quotients.

arXiv commentThe third version, updated. 27 pages, 1 figure