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Automorphisms of finite order on Gorenstein del Pezzo surfaces
Authors:D.-Q. Zhang
Affiliation:Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore
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 $vert-K_{Y}vert$. In particular, we show that either $vertoperatorname{Aut}(Y)vert = 2^{a}3^{b}$ 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 $vertoperatorname{Aut}(Y)vert$ is infinite).

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