Rank 2 vector bundles over \(\mathbb P^2(\mathbb C)\) whose sections vanish on points in general position
type: Item
instance of: scholarly article
title: Rank 2 vector bundles over \(\mathbb P^2(\mathbb C)\) whose sections vanish on points in general position
zbMATH Open document ID: 1330.14070
author: Alberto Alzati
author: Alfonso Tortora
published in: Revista Matemática Complutense
publication date: 2015-09-23T00:00:00Z
review text: A \(0\)-dimensional subscheme \(W\) of \(\mathbb{P}^2_{\mathbb C}\) is said to be in \textit{very general position} if each subscheme \(W' \subseteq W\) imposes \(\mathrm{lenght}(W')\) conditions to any complete linear system of plane curves. The paper under review deals with the problem of classification of rank two vector bundles \(E\) such that \(h^0(\mathbb{P}^2,E) \geq 2\) and such that the general section is vanishing in a \(0\)-dimensional scheme in very general position (ZGP bundles). The authors are able to find all the possible values of the Chern classes of such \(E\)'s and subsequently their stability (with some exceptions). The complete classification according to their second Chern class is presented (last part of Sect. 4). Afterwards they partially ask the problem of fitting a ZGP bundle \(E\) in a particular exact sequence which leads to constructions of several new examples of threefolds \(\mathbb{P}(E)\) of low degree (\(11\) in \(\mathbb{P}^{10}\), for instance, see Theorem 9).
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 6484513
zbMATH Keywords: rank 2 vector bundles
zbMATH Keywords: general position
zbMATH Keywords: very ampleness
reviewed by: Roberto Muñoz
MaRDI profile type: Publication
full work available at URL: s13163-015-0173-y
OpenAlex ID: W2322979939
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DOI: 10.1007/S13163-015-0173-Y
description: scientific article; zbMATH DE number 6484513