Stratification of the moduli space of four-gonal curves
type: Item
instance of: scholarly article
zbMATH Open document ID: 1304.14031
DOI: 10.1017/S001309151300062X
author: Michela Brundu
author: Gianni Sacchiero
publication date: 2014-09-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1204.6156
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 6349667
zbMATH Keywords: four-gonal curves
zbMATH Keywords: moduli space
zbMATH Keywords: ruled surfaces
MaRDI profile type: Publication
arXiv ID: 1204.6156
cites work: Footnotes to a paper of Beniamino Segre. (The number of \(g^1_d\)'s on a general \(d\)-gonal curve, and the unirationality of the Hurwitz spaces of 4-gonal and 5-gonal curves)
cites work: Weierstrass gap sequences and moduli varieties of trigonal curves
cites work: Syzygies of canonical curves and special linear series
cites work: On the varieties parametrinzing trigonal curves with assigned weierstrass points
cites work: The genus of space curves
cites work: Algebraic surfaces and holomorphic vector bundles
cites work: On the Singularities of Surfaces Ruled by Conics
cites work: On Rational Surfaces Ruled by Conics
OpenAlex ID: W2167992640
title: Stratification of the moduli space of four-gonal curves
published in: Proceedings of the Edinburgh Mathematical Society. Series II
review text: A \(\gamma\)-gonal curve is a smooth irreducible projective curve that admits a degree \(\gamma\) branched cover of \(\mathbb P^1\). The locus of \(\gamma\)-gonal curves \(\mathcal M_{g, \gamma}\) provides an important series of subvarieties of the moduli space of genus \(g\) curves \(\mathcal M_g\). (Caution: here the notation \(\mathcal M_{g, \gamma}\) does \textit{not} stand for the moduli space of pointed curves.) For instance, if \(d=2\), \(\mathcal M_{g, 2}\) is the locus of hyperelliptic curves. For \(d=3\), the canonical model of a trigonal curve \(X\) lies in a scroll surface, which is spanned by the trigonal divisors on \(X\). The (un)balanced type of the scroll surface defines the so-called Maroni invariant of \(X\). In particular, loci of trigonal curves with the same Maroni invariant yield a stratification of \(\mathcal M_{g, 3}\).NEWLINENEWLINEIn the paper under review, the authors study the locus of four-gonal curves \(\mathcal M_{g, 4}\). Analogous to the case of trigonal curves, they show that for a four-gonal curve \(X\), there (almost always uniquely) exists a surface, ruled by conics, containing the canonical model of \(X\). By analyzing the geometry of such a surface, the authors are able to extract several discrete invariants associated to \(X\). They further study the relation of those invariants as well as their geometric meanings and possible ranges. Finally, the authors apply the invariants to describe a stratification of \(\mathcal M_{g, 4}\). The existence of such a surface has also been obtained, using a different method, by \textit{F.-O. Schreyer} [Math. Ann. 275, 105--137 (1986; Zbl 0578.14002)].
reviewed by: Dawei Chen
description: scientific article; zbMATH DE number 6349667