On the restriction of vector bundles to good families of curves
type: Item
instance of: scholarly article
title: On the restriction of vector bundles to good families of curves
zbMATH Open document ID: 0835.14004
DOI: 10.1007/BF01760330
author: E. Ballico
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1996-04-15T00:00:00Z
review text: The author proves several results concerning vector bundles on projective manifolds by restricting them to suitable curves. In the first paragraph, he considers a threefold \(X\), the contraction of an extremal ray \(f : X \to Y\) and a vector bundle \(E\) on \(X\) and gives a criterion for \(E\) to be the pull-back of a vector bundle on \(Y\). In the rest of the paper, the author proves the semi-stability of some homogeneous vector bundles on \(\mathbb{P}^n\). Let us illustrate his method by his proof of the semi-stability of the kernel \(E\) of the evaluation map: \(H^0{\mathcal O}_\mathbb{P} (d)\otimes_k{\mathcal O}_\mathbb{P} \to{\mathcal O}_\mathbb{P} (d)\). According to some older results of the author and \textit{Ph. Ellia} [Math. Z. 196, 355-367 (1987; Zbl 0631.14029)] there exist non-special curves \(C\) in \(\mathbb{P}^n\) with \(H^0 \dot I_C (d) = 0\), \(H^1 \dot I_C (d) = 0\). It follows that \(H^0 (E |C) = 0\), \(H^1 (E |C) = 0\) and this easily implies that \(E |C\) is semistable, whence \(E\) is semistable.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 711493
zbMATH Keywords: vector bundles
zbMATH Keywords: extremal ray
zbMATH Keywords: semi-stability
reviewed by: Iustin Coandă
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01760330
OpenAlex ID: W2051207913
description: scientific article; zbMATH DE number 711493