On the Existence of Special Stable Spanned Vector Bundles on Projective Curves

Let X be a smooth projective curve of genus g ≥ 2 defined over an arbitrary algebraically closed field K. For any vector bundle E on X, call μ(E) := deg(E)/rank(E) the slope of E. A vector bundle E on X is said to be spanned if the natural map H(X,E)⊗OX → E is surjective. Such bundles are important tools for the projective geometry ofX because they are exactly the vector bundles associated to a morphism from X into a Grassmannian. A vector bundle E on X is said to be stable (resp. semistable) if for every proper subsheaf A of E we have μ(A) < μ(E) (resp. μ(A) ≤ μ(E)). In particular every line bundle is stable. A stable line bundle, E, on X is simple, i.e., every endomorphism of E is induced by the multiplication by some λ ∈ K. The stable vector bundles on X are the more interesting bundles on X from the point of view of moduli problems. The aim of this paper is the proof of the following result.