Quotients of del Pezzo surfaces

Abstract

Let $\mathbb{k}$ be any field of characteristic zero, $X$ be a del Pezzo surface and $G$ be a finite subgroup in $\mathrm{Aut}(X)$. In this paper we study when the quotient surface $X/G$ can be non-rational over $\mathbb{k}$. Obviously, if there are no smooth $\mathbb{k}$-points on $X/G$ then it is not $\mathbb{k}$-rational. Therefore under assumption that the set of smooth $\mathbb{k}$-points on $X/G$ is not empty we show that there are few possibilities for non-$\mathbb{k}$-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-$\mathbb{k}$-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 $\mathbb{k}$-rational if the $G$-invariant Picard number of $X$ is $1$. For the groups of order $3$ and $6$ we construct examples of both $\mathbb{k}$-rational and non-$\mathbb{k}$-rational quotients of both $\mathbb{k}$-rational and non-$\mathbb{k}$-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-$\mathbb{k}$-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of $\mathbb{k}$-rational surfaces, and obtain some corollaries concerning fields of invariants of $\mathbb{k}(x,y)$.

35 pages, 4 figures, 1 table. arXiv admin note: text overlap with arXiv:1709.02006