Determinantal varieties associated to rank two vector bundles on projective spaces and splitting theorems
type: Item
instance of: scholarly article
title: Determinantal varieties associated to rank two vector bundles on projective spaces and splitting theorems
zbMATH Open document ID: 0954.14035
author: Hideyasu Sumihiro
published in: Hiroshima Mathematical Journal
publication date: 2000-06-07T00:00:00Z
review text: The author constructs certain determinantal varieties associated to a rank two vector bundle on a projective variety \(S\) of dimension \(n\). These determinantal varieties have codimension \([n/2]\), the integral part of \(n/2\). By a careful analysis of these varieties, the author proves theorems about vector bundles on projective spaces. The analysis seems very interesting. Some of the applications though, we believe, are well known to the experts. At any rate some of them have never appeared before in print, to the best of our knowledge. For example, the fact that if \(E\) is a rank two vector bundle on \({\mathbb{P}}^n\) (\(n\geq 3\)) with \(\text{H}^1(\text{End} E)=0\), then it is split. It is easy to show that the condition implies that \(E\) is homogeneous and thus split if \(n\geq 4\). But the reviewer has not seen this fact written down before.
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 1347354
zbMATH Keywords: vector bundles
zbMATH Keywords: Hilbert schemes
zbMATH Keywords: determinantal varieties
reviewed by: N. Mohan Kumar
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206125016
OpenAlex ID: W1542111130
DOI: 10.32917/HMJ/1206125016
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description: scientific article; zbMATH DE number 1347354