Bound of automorphisms of surfaces of general type. I
type: Item
instance of: scholarly article
title: Bound of automorphisms of surfaces of general type. I
zbMATH Open document ID: 0811.14011
DOI: 10.2307/2946627
author: Gang Xiao
published in: Annals of Mathematics. Second Series
publication date: 1995-05-02T00:00:00Z
review text: A famous theorem of Hurwitz asserts that the number of automorphisms of a nonsingular projective complex curve \(X\) of genus \(g\geq 2\) is bounded by \(84 (g-1)= 42\deg K_ X\). In this paper this result extends to minimal algebraic surfaces of general type. The author proves that the number of automorphisms of such a surface \(X\) is bounded by \(CK_ X^ 2\) for some universal constant \(C\). The constant \(C\) can be taken equal to 288 unless \(X/\Aut (X)\) is a rational surface. In the forthcoming part II of this paper the author proves that \(C\) can be taken equal to \(42^ 2\) which gives the exact analog of the Hurwitz theorem. The history of previous attempts to prove this remarkable result is rather long. Until very recently no polynomial (in \(K_ X^ 2)\) bounds for the order of \(\Aut(K)\) was known. The first polynomial bounds were found by \textit{A. Corti} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 24, 113-137 (1991; Zbl 0758.14026)] and \textit{A. T. Huckleberry} and \textit{M. Sauer} [Math. Z. 205, No. 2, 321-329 (1990; Zbl 0718.32022)]. -- The proof of the theorem follows the idea of Hurwitz. One divides the surface by its full automorphism group and analyzes the finite cover \(X\to X/\Aut (X)\). The obvious difficulty here is that the quotient surface \(Y=X/ \Aut(X)\) could be singular. The author uses the technique of rational divisors to show that \(\#\Aut (X)/ K_ X\) is bounded by \(1/L^ 2\) where \(L\) is a rational divisor contained in the sum of the canonical class of \(Y\) and the branch divisor. Recently the result of the author has been extended to the case of surfaces of log-general type by \textit{V. Alexeev} [Int. J. Math. 5, No. 6, 779-810 (1994)].
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 14J17
zbMATH DE Number: 549322
zbMATH Keywords: number of automorphisms
zbMATH Keywords: minimal algebraic surfaces of general type
zbMATH Keywords: Hurwitz theorem
reviewed by: Igor V. Dolgachev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946627
OpenAlex ID: W2318887141
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description: scientific article; zbMATH DE number 549322