Bounding sections of bundles on curves

Abstract

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).