Variety of special linear systems on \(k\)-sheeted coverings
type: Item
instance of: scholarly article
title: Variety of special linear systems on \(k\)-sheeted coverings
zbMATH Open document ID: 0903.14009
DOI: 10.1023/A:1004928524154
author: Chang Ho Keem
author: Akira Ohbuchi
author: Takao Kato
published in: Geometriae Dedicata
publication date: 1999-01-03T00:00:00Z
review text: Let \(X\) be a smooth algebraic curve of genus \(g\) which admits a \(k\)-sheeted covering onto a general curve \(C\) of genus \(h > 0\), where \(k\) is an odd prime. In the first section, certain integers \(\varepsilon (h,k)\) and \(\delta (h,k)\) are defined. The authors prove that for \(g \geq \varepsilon(h,k)\), the variety \(W^1_d (X)\) of pencils of degree \(d\) on \(X\) is generically reduced and irreducible with the expected dimension, for all \(d \geq \delta(h,k)\). This result is sharp, in the sense that the authors also show that \(W^1_{\delta (h,k)-1}\) is not irreducible. The authors also discuss the validity of the theorem for \(k=2\) and they discuss sharpness for \(k=3\). In the second section the authors show that for suitable \(d\) and \(g\), and for a general choice of \(X\) as above, there exists an irreducible component of \(W^1_d (X)\) containing a base-point-free \(g^1_d\) which is not composed with the given \(k\)-sheeted covering. Furthermore, every such component is generically smooth of dimension \(\rho(d,g,1)\) (the Brill-Noether number, which is the expected dimension).
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14C21
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 1127654
zbMATH Keywords: algebraic curve
zbMATH Keywords: linear series
zbMATH Keywords: branched covering
zbMATH Keywords: genus
zbMATH Keywords: pencils
zbMATH Keywords: Brill-Noether number
reviewed by: Juan C. Migliore
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1127654