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Regular version of the site

Article

Quotients of del Pezzo surfaces

International Journal of Mathematics. 2019. Vol. 30. No. 11.

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