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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Automorphisms of finite order on Gorenstein del Pezzo surfaces


Author: D.-Q. Zhang
Journal: Trans. Amer. Math. Soc. 354 (2002), 4831-4845
MSC (2000): Primary 14J50; Secondary 14J26
Published electronically: August 1, 2002
MathSciNet review: 1926853
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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 $\vert\operatorname{Aut}(Y)\vert = 2^{a}3^{b}$ 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 $\vert\operatorname{Aut}(Y)\vert$ is infinite).


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Additional Information

D.-Q. Zhang
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore
Email: matzdq@math.nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03069-6
PII: S 0002-9947(02)03069-6
Received by editor(s): March 10, 2002
Published electronically: August 1, 2002
Article copyright: © Copyright 2002 American Mathematical Society