On the discriminant locus of a rank \(n-1\) vector bundle on \(\mathbb{P}^{n-1}\)
type: Item
instance of: scholarly article
title: On the discriminant locus of a rank \(n-1\) vector bundle on \(\mathbb{P}^{n-1}\)
zbMATH Open document ID: 1469.14034
DOI: 10.4171/PM/2053
author: Hirotachi Abo
published in: Portugaliae Mathematica. Nova Série
publication date: 2021-07-30T00:00:00Z
review text: The paper under review is devoted to study geometric properties of the discriminant locus of a vector bundle of rank \(n - 1\) on the projective \((n - 1)\)-space \(\mathbb P^{n-1}\). The discriminant locus \(\Delta(\mathcal E)\) of such a vector bundle \(\mathcal E\) is the subset of \(\mathbb P(H^0(\mathcal E))\) parameterizing the zero schemes of sections of \(\mathcal E\) which are either singular or positive dimensional. In many cases, \(\Delta(\mathcal E)\) is irreducible of codimension one, e.g.\ if \(\mathcal E\) is a direct sum of very ample line bundles and its top Chern class is greater than one. There are, however, indecomposable vector bundles whose discriminant loci are not irreducible or have higher codimensions. For linear systems of divisors, the problem of classifying degenerate discriminant loci was extensively studied in the case of very ample line bundles, while for more general line bundles its study was started by \textit{A. Lanteri} et al. [J. Reine Angew. Math. 477, 199--219, appendix: 216--219 (1996; Zbl 0851.14002)] and continued by \textit{A. Lanteri} and \textit{R. Muñoz} [J. Pure Appl. Algebra 212, No. 4, 808--831 (2008; Zbl 1152.14006); Indiana Univ. Math. J. 58, 2205--2225 (2009; Zbl 1205.14007)]; the classification, however, is still incomplete. As to vector bundles of higher rank, the classification problem has never systematically been explored. The paper under review highlights a large family of vector bundles whose discriminant loci are non-degenerate. More precisely, the main result the author proves, working over an algebraically closed field \(k\) of characteristic zero, is the following. If \(\mathcal E\) is a very ample vector bundle of rank \(n-1\) over \(\mathbb P^{n-1}\) with \(c_{n-1}(\mathcal E) \geq 2\), then \(\Delta(\mathcal E)\) is an irreducible hypersurface whose degree is expressed as an explicit function of invariants of \(\mathcal E\). A remarkable example is offered by Steiner bundles, roughly speaking very am…
Mathematics Subject Classification ID: 14F06
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 15A69
zbMATH DE Number: 7376803
zbMATH Keywords: vector bundle
zbMATH Keywords: discriminant variety
zbMATH Keywords: determinantal variety
reviewed by: Antonio Lanteri
describes a project that uses: Macaulay2
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/pm/2053
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description: scientific article; zbMATH DE number 7376803