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Bounding sections of bundles on curves



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Sols, Ignacio and Arrondo Esteban, Enrique (1992) Bounding sections of bundles on curves. In Complex projective geometry. London Mathematical Society Lecture Note Series (179). Cambridge University Press , Trieste, pp. 24-31. ISBN 0-521-43352-5

Official URL: http://www.cambridge.org/gb/knowledge/isbn/printView/item6532865/?site_locale=en_GB


The aim of this note is to prove some bounds on the global sections of vector bundles over a smooth, complete and connected curve C . Just by an application of the Clifford theorem, the authors prove (Proposition 2) (*) h 0 (E)≤deg(E)/2+2 for a semistable rank 2 vector bundle E and discuss when (*) is sharp. They propose a sharper bound for an indecomposable bundle (which is shown to be correct for a hyperelliptic curve) but, as added in proof, this bound is overoptimistic in the general case (see Proposition IV.7 of a paper by the reviewer [Duke Math. J. 64 (1991), no. 2, 333–347] or forthcoming work of Tan). By a dimension count the authors prove (Corollary 6) h 0 (E)≤deg(E)/2+rank(E) for every globally generated semistable bundle E . In this set-up, they give a Martens-type theorem (Proposition 9).

Item Type:Book Section
Uncontrolled Keywords:Ruled surface; Hyperelliptic curves; Non-semistable vector bundles
Subjects:Sciences > Mathematics > Algebra
ID Code:20987
Deposited On:22 Apr 2013 15:47
Last Modified:22 Jan 2016 15:09

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