Stability of tautological bundles on the Hilbert scheme of two points on a surface

Abstract

Let (X,H) be a polarized smooth projective surface satisfying H^1(X,O_X)=0 and let F be either a rank one torsion-free sheaf or a rank two μH-stable vector bundle on X. Assume that c_1(F)/=0. In this article it is shown that the rank two, respectively rank four tautological sheaf F^{[2]} associated with F on the Hilbert square X^{[2]} is μ-stable with respect to a certain polarization.

14 pages, minors corrections of typos, replaced Lemma 2.2 by Def 2.2 and Prop. 2.3, to appear in Nag. Math. J