On rationally determined line bundles on a family of projective curves with general moduli
type: Item
instance of: scholarly article
title: On rationally determined line bundles on a family of projective curves with general moduli
zbMATH Open document ID: 0657.14013
DOI: 10.1215/S0012-7094-87-05545-1
author: Ciro Ciliberto
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Fix integers g,r,d; set \(\rho:=\rho (g,r,d):=g-(r+1)(g+r-d)\). Let H be the open subset of \(Hilb({\mathbb{P}}^ r)\) parametrizing smooth, irreducible, nondegenerate curves in \({\mathbb{P}}^ r\) with degree d and genus g. Let \(t:\quad F\to H\) be the universal curve. For every Zariski open subset U of H, set \(R(F_ U)=Co\ker (t^*:Pic(U)\to Pic(t^{- 1}(U))\); \(R(F_ U)\) is the group of rationally determined line bundles over U. Here it is proved (using a theorem of Harer) that if \(\rho\geq 2\), for every \(U\neq \emptyset \quad R(F_ U)\) is generated by the class of the relative canonical line bundle and the class of the hyperplane bundle. It would be very interesting to settle the cases \(\rho =0\) and \(\rho =1\), as well as to solve the corresponding problem for other components with \(\rho <0\).
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14C05
zbMATH DE Number: 4073154
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: Picard group
zbMATH Keywords: moduli scheme
zbMATH Keywords: degree
zbMATH Keywords: genus
zbMATH Keywords: universal curve
zbMATH Keywords: canonical line bundle
zbMATH Keywords: hyperplane bundle
MaRDI profile type: Publication
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full work available at URL: s0012-7094-87-05545-1
OpenAlex ID: W1492052085
description: scientific article; zbMATH DE number 4073154