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 in \cite{OG1} 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 $v^{\perp}$ (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 $M_{10}$. Similar results are shown for the moduli space $M_{6}$ introduced by O'Grady in \cite{OG2}.