On vector bundles over surfaces and Hilbert schemes

Let X be an irreducible smooth projective surface over and Hilb (d) (X) the Hilbert scheme parametrizing the zero-dimensional subschemes of X of length d. Given a vector bundle E on X, there is a naturally associated vector bundle over Hilb (d) (X). If E and V are semistable vector bundles on X such that and are isomorphic, we prove that E is isomorphic to V. A key input in the proof is provided by Biswas and Nagaraj (see [1]).