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Generalization of a criterion for semistable vector bundles

Indranil Biswas, Georg Hein

Number 27
Author Georg Hein
Project C10
Year 2008

It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that both H0(X,EF) and H1(X,EF) vanish. We extend this criterion
for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on X such that Hi(X,EF)=0 for all i. We also give an explicit bound for the rank of F.

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