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 $H^1(X,{\mathcal{O}}_X)=0$, and let $\mathcal{F}$ be either a rank 1 torsion-free sheaf or a rank 2 ${\mu }_H$-stable vector bundle on $X$. Assume that $c_1\left(\mathcal{F}\right)\ne 0$. This article shows that the rank 2—respectively, rank 4—tautological sheaf ${\mathcal{F}}^{\left[2\right]}$ associated with $\mathcal{F}$ on the Hilbert square $X^{\left[2\right]}$ is $\mu$-stable with respect to a certain polarization.