Globally generated fibre bundles over the projective space
type: Item
instance of: scholarly article
title: Globally generated fibre bundles over the projective space
zbMATH Open document ID: 0821.14009
DOI: 10.1007/BF01446640
author: Laurent Manivel
published in: Mathematische Annalen
publication date: 1995-10-03T00:00:00Z
full work available at URL: https://eudml.org/doc/165302
review text: \textit{P. Ellia} and \textit{A. Hirschowitz} recently [cf. J. Algebr. Geom. 1, No. 4, 531-547 (1992; Zbl 0812.14036)] suggested that a ``general'' fiber bundle of rank \(r\) on a smooth projective \(n\)-dimensional variety should be globally generated as soon as it possesses \(n + r\) independent global sections. This principle would provide a useful complement to the classical Castelnuovo-Mumford criterion. We prove a conjecture of \textit{A. Dolcetti} [Math. Ann. 294, No. 1, 99-107 (1992; Zbl 0782.14017)] that this principle holds true for vector bundles on \(\mathbb{P}^ n\) constructed as kernels of generic morphisms from \({\mathcal O}_{\mathbb{P}^ n} (d)^{\oplus b}\) to \({\mathcal O}_{\mathbb{P}^ n} (d + 1)^{\oplus a}\). These bundles are twisted duals of Steiner bundles, following the terminology of Dolgachev and Kapranov. Our proof relies on a careful dimension count for the exceptional sets of such morphisms.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 759278
zbMATH Keywords: global generation of fiber bundle
MaRDI profile type: Publication
cites work: Q5607044
cites work: Arrangements of hyperplanes and vector bundles on \(\mathbb{P}^ n\)
cites work: On the generation of certain bundles over \(\mathbb{P}^ 3\)
cites work: Q5288186
cites work: THE LINEAR SYZYGIES OF GENERIC FORMS
cites work: Q4128900
description: scientific article; zbMATH DE number 759278