Abstract
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 (Kähler) 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).