Space models of degree \(g+1\) of curves of genus \(g\)
type: Item
instance of: scholarly article
title: Space models of degree \(g+1\) of curves of genus \(g\)
zbMATH Open document ID: 1268.14031
author: Chang Ho Keem
author: Gerriet Martens
author: Marc Coppens
published in: Manuscripta Mathematica
publication date: 2013-02-25T00:00:00Z
review text: Let \(X\) be a smooth irreducible curve of genus \(g>1\) over the complex field. In the article under review, the authors consider the problem of the existence of a smooth linearly normal model of degree \(g+1\) of \(X\) in \(\mathbb P^3\), i.e., of a complete and very ample linear series \(g^3_{g+1}\) on \(X\). Note that, by Riemann-Roch, the degree \(d\) of a complete linear series of dimension \(3\) on \(X\) is at most \(g+3\), and that the condition \(d\geq 3/4g+3\) is required in order that the Brill-Noether number \(\rho(g,d,3)\) is non-negative. The existence of a complete and very ample \(g^3_{g+3}\) on any curve was proved by Halphen in 1882. The case \(d=g+2\) was treated by \textit{M. Coppens} and \textit{G. Martens} in [Math. Nachr. 213, 35--55 (2000; Zbl 0972.14021)]. Here the authors give a complete answer for \(d=g+1\) in the more general situation of a complete, base-point-free and simple linear series. The main theorem is the following: If \(g\geq 8\), then \(X\) has a complete base-point-free and simple \(g^3_{g+1}\) if and only if \(X\) is not hyperelliptic, not bielliptic and not a smooth plane sextic. The authors give an explicit example of a \(6\)-gonal curve of genus \(10\) not having any complete and very ample \(g^3_{11}\). As for complete and very ample \(g^3_{g+1}\)'s, such a linear series is proved to exist either if \(g\geq 8\), the gonality is different from \(3\) and \(\dim W^2_{k+5}\leq 2\), or if \(X\) is any smooth plane curve of degree \(d\geq 10\). Finally the following generalization of the main theorem is stated: if \(r\geq 2\) and \(\rho(g,g+r-2,r)\geq 0\), then there exists a complete base-point-free simple \(g^r_{g+r-2}\) if and only if \(X\) is not hyperelliptic, not bielliptic and not a smooth plane curve of degree at most \(10\).
reviewed by: Emilia Mezzetti
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14H51
zbMATH DE Number: 6138666
zbMATH Keywords: linear series
zbMATH Keywords: gonality
zbMATH Keywords: index of speciality
zbMATH Keywords: hyperelliptic
zbMATH Keywords: bielliptic
MaRDI profile type: Publication
full work available at URL: s00229-012-0539-0
OpenAlex ID: W2035442666
cites work: Plane Models for Riemann Surfaces Admitting Certain Half-Canonical Linear Series. II
cites work: Q3347989
cites work: Q4324882
cites work: Embeddings of general blowing-ups at points.
cites work: Non-trivial Linear Systems on Smooth Plane Curves
cites work: Primitive linear series on curves
cites work: Q4954309
cites work: A remark on very ample linear series
cites work: Teilkörper höheren Geschlechts eines algebraischen Funktionenkörpers
DOI: 10.1007/S00229-012-0539-0
description: scientific article; zbMATH DE number 6138666