Nodal cubic surfaces and the rationality of the moduli space of curves of genus two
type: Item
instance of: scholarly article
title: Nodal cubic surfaces and the rationality of the moduli space of curves of genus two
zbMATH Open document ID: 0541.14040
DOI: 10.1007/BF01455306
author: Fabio Bardelli
author: A. Del Centina
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/182982
review text: Let \({\mathcal M}_ 2\) be the moduli space of curves of genus 2 and \({\mathcal S}\) be the moduli space of ''nodal cubics'' i.e. of cubic surfaces with one ordinary double point. We produce a birational map \({\mathcal S}\to {\mathcal M}_ 2\) as follows: to a nodal cubic S with note P we associate a genus 2 curve \(\Gamma_ S\) which is the double cover of the conic \({\mathbb{P}}roj(\tan gent\quad cone\quad to\quad S\quad at\quad P)\) branched at the six points which are the lines through P lying on S; conversely, the bicanonical image of a genus 2 curve \(\Gamma\) is a conic in \({\mathbb{P}}^ 2\) with six distinguished points on it and this determines a nodal cubic. By using the classical Sylvester's theorem which gives a unique ''canonical'' equation for a generic (even nodal) cubic surface we construct an explicit ''birational model'' W for \({\mathcal S}\). W is a hypersurface of \({\mathbb{P}}^ 4/\sigma_ 5\) where \(\sigma_ 5\) is the symmetric group on 5 letters. The rationality of W is then proved by some computations, thus yielding the rationality of \({\mathcal S}\) and \({\mathcal M}_ 2\) over an algebraically closed field of characteristic 0.
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 3859311
zbMATH Keywords: rationality of moduli space of curves of genus 2
zbMATH Keywords: rationality of moduli space of nodal cubics
MaRDI profile type: Publication
cites work: Q3858373
cites work: Arithmetic variety of moduli for genus two
cites work: Q5525818
description: scientific article; zbMATH DE number 3859311