Quotient curves of smooth plane curves with automorphisms
type: Item
instance of: scholarly article
title: Quotient curves of smooth plane curves with automorphisms
zbMATH Open document ID: 1192.14028
author: Takeshi Harui
author: Takao Kato
author: Jiryo Komeda
author: Akira Ohbuchi
published in: Kodai Mathematical Journal
publication date: 2010-05-17T00:00:00Z
review text: Let \(C\) be a smooth plane curve over \({\mathbb C}\) of degree \(d\geq 4\) and suppose that \(\sigma\) is an automorphism of \(C\) of order \(n\geq 2\) with number of fixed points equal to \(f\). Then \(\sigma\) extends to an automorphism \(\tilde{\sigma}\) of \({\mathbb P}^2\) which can be represented by a \(3\times 3\) diagonal matrix with either two \(1\)'s on the diagonal (called a type 1 automorphism) or a single \(1\) on the diagonal (called a type 2 automorphism). In the paper under review, the authors determine some properties of the quotient curve \(B=C/\langle \sigma \rangle\), the automorphism \(\sigma\) and the degree \(d\) for type 1 and 2 automorphisms. For type \(1\) automorphisms, the authors prove a number of extensive results which hold for any \(n\). For example, the authors show that \(d\equiv 0\) or \(1\mod{(n)}\) and moreover when \(d\equiv 0\mod{(n)}\) we have \(f=d\), and when \(d\equiv 1\mod{(n)}\), \(f=d+1\). Additionally they show that the quotient curve \(B\) is isomorphic to a curve of degree \(d\) in \({\mathbb P}^{n+1}\) attaining the largest geometric genus given by the Castelnuovo bound. They also prove a converse statement. Specifically, given a curve \(B\) satisfying the properties derived, and integers \(n\geq 2\) and \(d\equiv 0\) or \(1\mod{(n)}\), there always exists a smooth plane curve \(C\) and an automorphism \(\sigma\) of order \(n\) of type 1 and a cyclic covering \(\pi : C\rightarrow B=C/\langle \sigma \rangle\). For type 2 automorphisms, the authors are able to provide similar partial results in the special case that \(n=p\) a prime. Specifically, they show that either \(f=0\) and \(d\equiv 0\mod{(p})\), \(f=2\) and \(d\equiv 1\) or \(2\mod{(p})\), or \(f=3\) and \(d^2-3d+3\equiv 0\mod{(p})\) and \(p\equiv 1\mod{(6)}\) or \(p=3\). They finish by giving specific examples to show that each of these three cases occur.
reviewed by: Aaron Wootton
Mathematics Subject Classification ID: 14H51
zbMATH DE Number: 5708833
zbMATH Keywords: plane curves
zbMATH Keywords: automorphisms
zbMATH Keywords: covering
zbMATH Keywords: quotient curves
zbMATH Keywords: linear system
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1270559164
OpenAlex ID: W2072365522
DOI: 10.2996/KMJ/1270559164
description: scientific article; zbMATH DE number 5708833