Segre classes of tautological bundles on Hilbert schemes of surfaces

We first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande on top Segre classes of the tautological bundles on Hilbert schemes of K3 surfaces equipped with a line bundle. We then turn to the blow-up of a K3 surface at one point and establish vanishing results for the corresponding top Segre classes in a certain range. This determines, at least theoretically, all top Segre classes of tautological bundles for any pair (Sigma, H), where H is an element of Pic Sigma.