The 2-factoriality of the O'Grady moduli spaces

The aim of this work is to show that the moduli space M_{10} introduced by O'Grady is a 2-factorial variety. Namely, M_{10} is the moduli space of semistable sheaves with Mukai vector v:=(2,0,-2)\in H^{ev}(X,\mathbb{Z}) on a projective K3 surface X. As a corollary to our construction, we show that the Donaldson morphism gives a Hodge isometry between v (sublattice of the Mukai lattice of X) and its image in H^{2}(\widetilde{M}_{10},\mathbb{Z}), lattice with respect to the Beauville form of the 10-dimensional irreducible symplectic manifold \widetilde{M}_{10}, obtained as symplectic resolution of M_{10}. Similar results are shown for the moduli space M_{6} introduced by O'Grady to produce its 6-dimensional example of irreducible symplectic variety. © Springer-Verlag 2009.