Deforming torison-free sheaves on an algebraic surface

This paper investigates the question of removability of singularities of torsion-free sheaves on algebraic surfaces in the universal deformation and the existence in it of a nonempty open set of locally free sheaves, and describes the tangent cone to the set of sheaves having degree of singularities larger than a given one. These results are used to prove that quasitrivial sheaves $\mathscr F$ on an algebraic surface $X$ with $c_2(\mathscr F)>(r+1)\max(1,p_g(X))$ have a universal deformation whose general sheaf is locally free and stable relative to any ample divisor on $X$, and thereby to find a nonempty component of the moduli space of stable bundles on $X$ with $c_1=0$ and $c_2>\max(1,p_g(X))\cdot(r+1)$ on any algebraic surface.