Höhere Sekantenvarietäten und Vektorbündel auf Kurven. (Higher secant varieties and vector bundles on curves)
type: Item
instance of: scholarly article
title: Höhere Sekantenvarietäten und Vektorbündel auf Kurven. (Higher secant varieties and vector bundles on curves)
zbMATH Open document ID: 0588.14030
DOI: 10.1007/BF01171486
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155078
review text: The main purpose of the article is to prove the following nice result: Let E be a locally free sheaf of rank 2 on a non-singular irreducible curve of genus g defined over an algebraically closed field of characteristic zero. An invertible (split) subsheaf M of E is called ''maximal'' if \(c_ 1(M)\) is maximal. Let \(s=c_ 1(E)-2c_ 1(M)\) for M maximal. Assume that the following three properties hold: \((1)\quad 2\leq s\leq g-1;\) \((2)\quad h^ 0(X,\det (E\otimes M^{-1}))\leq 1\) for a maximal M; \((3)\quad E\) has only a finite number of maximal subbundles. - Then we have that the number of maximal subbundles is at most equal to \(\sum^{s}_{k=0}\left( \begin{matrix} g-1-k\\ s-k\end{matrix} \right)\left( \begin{matrix} g\\ k\end{matrix} \right)+1.\) The proof uses results and techniques of the author and \textit{M. S. Narasimhan} [Math. Ann. 266, 55-72 (1983; Zbl 0507.14005)] to relate the above result to results about s-secant (s-1)-planes. Such secant planes are studied using Schwarzenberger's theory of secant planes to curves. The latter theory is reviewed in the article.
Mathematics Subject Classification ID: 14N10
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14H99
zbMATH DE Number: 3944031
zbMATH Keywords: number of maximal subbundles
zbMATH Keywords: secant planes to curves
author: Herbert Lange
reviewed by: Dan Laksov
MaRDI profile type: Publication
cites work: Q3328651
cites work: Higher secant varieties of curves and the theorem of Nagata on ruled surfaces
cites work: Secant Bundles on Symmetric Products
cites work: The Secant Bundle of a Projective Variety
OpenAlex ID: W2914254366
description: scientific article; zbMATH DE number 3944031