Brill-Noether theory for rank two vector bundles generated by their sections
type: Item
instance of: scholarly article
title: Brill-Noether theory for rank two vector bundles generated by their sections
zbMATH Open document ID: 1166.14012
author: Jianbo Xiao
author: Xiao-Jiang Tan
publication date: 2009-01-05T00:00:00Z
review text: The paper under review is a contribution to the Brill-Noether theory of vector bundles on algebraic curves [see \textit{I. Grzegorczyk and M. Teixidor I. Bigas}, \url{arXiv:0801.4740}] for an overview of the main known results about stable vector bundles. Let \(C\) be a reduced irreducible smooth non-hyperelliptic complex projective algebraic curve of genus \(g \geq 3\). A rank two vector bundle \(E\) on \(C\) is said to be \textit{generated by its sections} if \(E\) is an extension \(0 \to L_1 \to E \to L_2 \to 0\) of a line bundle \(L_2\) by a line bundle \(L_1\) such that \(H^0 (C,L_1) \neq 0\) and \(Im \{ H^0 (C,E) \to H^0 (C,L_2) \} \neq 0\). Recall also that a bundle \(E\) is called \textit{special} if both \(H^0 (C, E)\) and \(H^1 (C, E)\) are nonzero. The authors first construct a holomorphic vector bundle \(H_d\) of dimension \(d\) on \(C_d\) such that each point \((D, f) \in H_d\), \(D \in C_d\) and \(f \in H^0 (C, {O_C}_{|D})\), uniquely corresponds to a base point free rank two vector bundle \(E\) generated by its sections. Thus \(H_d\) can be regarded as a parameter space of all base point free rank two vector bundles of degree \(d\) on \(C\) generated by their sections. Then they obtain a condition for splitting of \(E\) in terms of the decomposition of \(D\) as a sum of effective divisors. Further, they prove that if \(E\) is indecomposable, generated by its sections, and \(d > 6g - 6\) then \(H^1 (C, E) = 0\), while in case \(d = 6g - 6\) there exists a unique \(E\) with these properties and such that \(H^1 (C, E) \neq 0\). Next, they produce two Clifford-like results according to whether the bundle splits, i.e. they find upper bounds for \(\dim H^0 (C, E)\) in terms of its degree and the genus \(g\). Finally, the authors prove that the existence of rank two vector bundles on a general curve depends on the non-negativity of the Brill-Noether number \(\rho (g,r,d) = g - (r+1)(g-d+r)\). Precisely, denote by \(H^r_d = \{ E \in H_d | \dim H^0 (C,E) \geq …
reviewed by: Hristo Krassimirov Iliev
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14H60
zbMATH DE Number: 5485344
zbMATH Keywords: rank two vector bundle
zbMATH Keywords: vanishing theorem
zbMATH Keywords: Clifford theorem
zbMATH Keywords: existence theorem
MaRDI profile type: Publication
full work available at URL: s10114-008-7268-z
OpenAlex ID: W1977692751
cites work: Q3347989
cites work: On the variety of special linear systems on a general algebraic curve
cites work: On the Gieseker-Petri map for rank 2 vector bundles
cites work: Some results on the existence of rank two special stable vector bundles
DOI: 10.1007/S10114-008-7268-Z
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5485344