On the Symmetric Algebra of Stable Vector Bundles on Curves

Abstract

Let X be a smooth projective curve of genus g ≥ 2 and E a general stable vector bundle on X with rank(E) = r and deg(E) = d > 0. Let z be the smallest integer with zd > 2gr. Here we study the graded algebra H0 (X, Sym(E)) := ⌖n≥0H0 (X, Sn (E)) and prove that it is generated by elements of degree ≤ z.