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

TL;DR

This paper proves the μ-stability of certain tautological sheaves on the Hilbert scheme of two points on a surface, under specific conditions on the surface and the sheaves, contributing to the understanding of stability in algebraic geometry.

Contribution

It establishes the μ-stability of rank two and four tautological sheaves on the Hilbert square of a surface, extending stability results to new classes of sheaves.

Findings

Rank two tautological sheaves are μ-stable on X^{[2]}.

Rank four tautological sheaves are μ-stable on X^{[2]}.

Stability holds under specific polarization and surface conditions.

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 {\mu}H-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 {\mu}-stable with respect to a certain polarization.