Desingularized moduli spaces of sheaves on a K3, I

Moduli spaces of semistable torsion-free sheaves on a K3 surface $X$ are often holomorphic symplectic varieties, deformation equivalent to a Hilbert scheme parametrizing zero-dimensional subschemes of $X$. In fact this should hold whenever semistability is equivalent to stability. In this paper we study a typical "opposite" case, i.e. the moduli space $M_c$ of semistable rank-two torsion-free sheaves on $X$ with trivial determinant and second Chern class equal to an even number $c$. The moduli space $M_c$ always contains points corresponding to strictly semistable sheaves. If $c$ is at least 4, then $M_c$ is singular along the locus parametrizing strictly semistable sheaves, and on the smooth locus of $M_c$ there is a symplectic holomorphic form. Thus it is natural to ask whether there is a symplectic desingularization of $M_c$. We construct such a desingularization for $c=4$; in another paper we show that this desingularization gives a new deformation class of (K\"ahler) holomorphic irreducible symplectic varieties (of dimension ten). We also study the case $c>4$. We describe what should be an interesting desingularization, however we are not able to produce a symplectic one. In fact we suspect there is no symplectic smooth model of $M_c$ if $c>4$ (and even, of course).