Automorphisms of finite order on Gorenstein del Pezzo surfaces

Abstract:

In this paper we shall determine all actions of groups of prime order $p$ with $p \ge 5$ on Gorenstein del Pezzo (singular) surfaces $Y$ of Picard number 1. We show that every order-$p$ element in $\operatorname {Aut}(Y)$ ($= \operatorname {Aut}({\widetilde Y})$, ${\widetilde Y}$ being the minimal resolution of $Y$) is lifted from a projective transformation of ${\mathbf {P}}^{2}$. We also determine when $\operatorname {Aut}(Y)$ is finite in terms of $K_{Y}^{2}$, $\operatorname {Sing} Y$ and the number of singular members in $|-K_{Y}|$. In particular, we show that either $|\operatorname {Aut}(Y)| = 2^{a}3^{b}$ for some $1 \le a+b \le 7$, or for every prime $p \ge 5$, there is at least one element $g_{p}$ of order $p$ in $\operatorname {Aut}(Y)$ (hence $|\operatorname {Aut}(Y)|$ is infinite).