On the locus of curves with automorphisms
type: Item
instance of: scholarly article
title: On the locus of curves with automorphisms
zbMATH Open document ID: 0649.14013
DOI: 10.1007/BF01773930
author: Maurizio Cornalba
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1987-01-01T00:00:00Z
review text: Let \(S_ g\) be the locus of curves with a nontrivial automorphism in the moduli space \(M_ g\) of curves of genus \(g\geq 3\) (for \(g\geq 4\), \(S_ g\) is also the singular locus of \(M_ g)\). The author describes the irreducible components of \(S_ g\) in the following way: if C is a smooth curve of genus \(g\) with an automorphism \(\tau\) of prime order p, the quotient curve \(X=C/<\tau >\) is smooth of genus g', and \(C\to X\) is ramified at \(n\geq 0\) points \(q_ 1,...,q_ n\) in C, such that the Riemann-Hurwitz relation \(2g-2=p\cdot (2g'-2)+n(p-1)\) is satisfied. If \(z_ i\) is a local parameter at \(q_ i\) and \(\zeta\) a primitive p-th root of unity, then there are unique integers \(a_ i\leq p-1\) such that \(z_ i(P)=\zeta^{a_ i}z_ i(\tau (P))\). The space \(S_ g(p,g';a_ 1,...,a_ n)\) of all such coverings is an irreducible subspace of \(M_ g\) of dimension \(3g'-3+n\), and with a few exceptions which are explicitly described in the text, all of them are irreducible components of \(S_ g\), and conversely each component of \(S_ g\) is of the form \(S_ g(p,g';a_ 1,...,a_ n).\) The main technical lemma asserts that in the situation above, Aut(C/X) is a normal subgroup of Aut(C) (with exceptions \(g'=0\), \(n=3\) or \(g'=1\), \(n=2)\); its proof needs some deformation theory. The paper concludes with a table of the irreducible components of \(S_ g\) for \(g\leq 50\).
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14D22
zbMATH DE Number: 4059568
zbMATH Keywords: automorphisms of curves
zbMATH Keywords: branched coverings
zbMATH Keywords: singular locus
zbMATH Keywords: moduli space of curves
zbMATH Keywords: deformation theory
reviewed by: Frank Herrlich
MaRDI profile type: Publication
cites work: Prym varieties and Schottky problem
cites work: The irreducibility of the space of curves of a given genus
cites work: On the Kodaira dimension of the moduli space of curves
cites work: Arithmetic variety of moduli for genus two
cites work: Q3712472
full work available at URL: https://doi.org/10.1007/bf01773930
OpenAlex ID: W4241195498
description: scientific article; zbMATH DE number 4059568