Bounding automorphism groups
type: Item
instance of: scholarly article
title: Bounding automorphism groups
zbMATH Open document ID: 0845.14009
DOI: 10.1007/BF01446320
author: Endre Szabó
published in: Mathematische Annalen
publication date: 1996-07-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165430
review text: It is an old problem to find an upper bound on the size of the group of birational automorphisms of a complex variety \(X\). It is known that this group is finite when \(X\) is of general type. In dimension 1 and 2 there is an upper bound which is linear in terms of Chern numbers of the minimal model of \(X\), and there is some (much larger) upper bound in all dimensions. Here we try to get a polynomial bound for \(n\)-dimensional \(X\). We have to assume that \(X\) has a minimal model \(Y\) and besides \(K_Y\) our bound will depend on the index \(r\) of \(Y\) as well. The bound is a polynomial of \((rK^n_Y)\). In the course of the proof it turns out to be useful to get a bound on the number of linear automorphisms of an arbitrary projective variety whenever this number is finite. This bound is a polynomial of the degree of the variety. We get some (much weaker) results in characteristic \(p > 0\). In the projective setup our bound is a polynomial of the degree and the size of the \(p\)-Sylow subgroup, this can be used to get some bounds for surfaces of general type (involving the size of the \(p\)-Sylow subgroup).
Mathematics Subject Classification ID: 14E07
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 14H37
zbMATH DE Number: 892575
zbMATH Keywords: group of birational automorphisms
zbMATH Keywords: Chern numbers
zbMATH Keywords: minimal model
zbMATH Keywords: surfaces of general type
MaRDI profile type: Publication
cites work: Q5801046
cites work: Q4855217
cites work: Polynomial bounds for the number of automorphisms of a surface of general type
cites work: A numerical criterion for very ample line bundles
cites work: Canonical models of surfaces of general type in positive characteristic
cites work: Q3328651
cites work: On the order of the automorphism group of a surface of general type
cites work: On the group of automorphisms of a function field
cites work: A note on abelian automorphism groups of surfaces of general type
cites work: Effective base point freeness
cites work: Q5344661
cites work: On Abelian Automorphism Groups of Algebraic Curves
cites work: Q2785567
cites work: Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I: Eine Abschätzung der Ordnung der Automorphismengruppe
cites work: On abelian automorphism group of a surface of general type
cites work: Bound of automorphisms of surfaces of general type. I
cites work: Q4859532
OpenAlex ID: W2159393521
description: scientific article; zbMATH DE number 892575