Automorphisms of finite order on Gorenstein del Pezzo surfaces

2002/02/14 by D. -Q. Zhang, Zhang, D. -Q.
Mathematics · #14J26 #14J50 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14J26 #msc:14J50

paper · pdf · doi:10.48550/arxiv.math/0202131

18 pages, Transactions of the American Mathematical Society, to appear

arxiv created 2002/02/14 · arxiv updated 2016/09/07

Abstract

In this note we shall determine all actions of groups of prime order p with p > 3 on Gorenstein del Pezzo (singular) surfaces Y of Picard number 1. We show that every order-p element in Aut(Y) (= Aut(Y'), Y' being the minimal resolution of Y) is lifted from a projective transformation of the projective plane. We also determine when Aut(Y) is finite in terms of the self intersection of the canonical divisor of Y, Sing(Y) and the number of singular members in the anti-canonical linear system of Y. In particular, we show that either |Aut(Y)| = 2^a 3^b for some 0 < a+b < 8, or for every prime p > 3 there is at least one element g_p of order p in Aut(Y) (hence |Aut(Y)| is infinite).

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