Unfiltrable vector bundles and spanned vector bundles

Abstract

Let X be an integral n-dimensional projective variety. We study the existence of rank r ≥ 2 vector bundles on X which are not extensions of two vector bundles (e.g. they exist if r=2 and X is a smooth surface with Kodaira dimension ≥ 0). Their existence implies the existence of many spanned rank n vector bundles E on X and (n+r)-dimensional linear subspaces W ⊆ H⁰(E) spanning E such that the evaluation map W⊗ O_X → E does not factor through f: W⊗ O_X → _F and g: F → E with F locally free, f, g surjective and n+r > rank(F) > rank(E).