A counter example of Arrondo-Sols' conjecture
type: Item
instance of: scholarly article
title: A counter example of Arrondo-Sols' conjecture
zbMATH Open document ID: 0784.14011
author: Xiao-Jiang Tan
published in: Chinese Science Bulletin
publication date: 1993-02-22T00:00:00Z
review text: Arrondo-Sol's conjecture can be stated as follows: Let \(E\) be a rank-2 vector bundle of degree \(d\) on a smooth complex algebraic curve of genus \(g\), \(-e\) be the minimal self-intersection number of a unisecant curve in the ruled surface \(p(E)\), and \(r+1=h^ 0(E)\). If \(-e \leq d \leq 4g-4+e\) and \(E \neq L \oplus L\) \((L\) line subbundle), then \(r \leq(d+e)/2\) [cf. \textit{E. Arrondo} and \textit{I. Sols} in Complex projective geometry, Sel. Pap. Conf. Proj. Var., Trieste 1989, and Vector Bundles and Special Proj. Embeddings, Bergen 1989, Lond. Math. Soc. Lect. Note Ser. 179, 24-31 (1992; Zbl 0772.14012)]. -- If the curve is hyperelliptic, then the conjecture is true (loc. cit.). The paper under review shows, by an example, that the conjecture is false for nonhyperelliptic curves of genus 3.
reviewed by: Paolo Valabrega
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 109357
zbMATH Keywords: hyperelliptic curve
zbMATH Keywords: rank-2 vector bundle
zbMATH Keywords: self-intersection number
zbMATH Keywords: curve in a ruled surface
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 109357