The 2-factoriality of the O’Grady moduli spaces

Abstract

The aim of this work is to show that the moduli space M10 introduced by O’Grady is a 2-factorial variety. Namely, M10 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 \({v^{\perp}}\) (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 M10. Similar results are shown for the moduli space M6 introduced by O’Grady to produce its 6-dimensional example of irreducible symplectic variety.