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Stable bundles of rank 2 with four sections. (English) Zbl 1237.14041

Let \(C\) be a smooth irreducible projective algebraic curve of genus \(g\geq 2\) defined over over \(\mathbb C\), \(M(n,d)\) (resp., \(\widetilde M(n,d)\)) be the moduli space of stable vector bundles (resp., S-equivalence classes of semistable vector bundles) of rank \(n\) and degree \(d\) over \(C\). The purpose of the article is to contribute to the study of the question of nonemptyness of Brill-Noether loci \(B(n,d,k)\) (resp., \(\widetilde B(n,d,k)\)) for \(n=2,\) \(k=4\). A lower bound on degree for nonempty Brill-Noether loci is known. The main result of the paper is a geometric criterion when the boundary is attained. The authors show that for a general curve of genus 10 the bound cannot be attained but there exist Petri curves if this genus for which the bound is sharp. Main results are interpreted for various curves and in terms of Clifford indices and coherent systems. The methods of the article provide a useful tool for the study of moduli of curves.

MSC:

14H60 Vector bundles on curves and their moduli
14H51 Special divisors on curves (gonality, Brill-Noether theory)
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