Construction of vector bundles and reflexive sheaves
type: Item
instance of: scholarly article
title: Construction of vector bundles and reflexive sheaves
zbMATH Open document ID: 0731.14004
author: H. Maeda
published in: Tokyo Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: This paper studies the relation between vector bundles and codimension 2 Cohen-Macaulay closed subschemes generalising the results of \textit{Vogelaar} [Thesis (Leiden 1978)]. Let X be a smooth algebraic variety over an (arbitrary) field k. Fix a vector bundle E of rank \(r-1\) and line bundles L, M, \(M=\det (E)\) on X. Given a vector bundle F of rank \(r,\det (F)=L\) and a sheaf injection h: \(E\to F\) such that the set Z of points where h is not of rank \(r-1\) is of pure codimension 2; the author shows that h determines a closed Cohen-Macaulay subscheme Y (with underlying set Z) and an element \(t_ Y\in H^ 0(E\otimes \omega_ Y(M-L-K_ X))\). The main result is that the correspondence f defined by \(f(F,h)=(Y,t_ Y)\) is surjective if \(H^ 2(E(M-L))=0\) and bijective if further \(H^ 1(E(M-L))=0\). The surjectivity of f is used to construct a new vector bundle of rank 3 on \({\mathbb{P}}^ 3\) using a rational singular curve which is not a locally complete intersection. Finally the author describes a way to construct reflexive sheaves on locally factorial Gorenstein varieties of dimension \(\geq 3\) using line bundles and codimension 2 closed subschemes.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14M07
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 4208244
zbMATH Keywords: vector bundles
zbMATH Keywords: closed Cohen-Macaulay subscheme
reviewed by: Usha N. Bhosle
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270133011
OpenAlex ID: W1985646667
DOI: 10.3836/TJM/1270133011
description: scientific article; zbMATH DE number 4208244