The 2-factoriaity of the O'Grady's moduli spaces of sheaves

Abstract.
In this talk, we study the two moduli spaces M₁₀ and M₆ introduced by O'Grady, which he uses to obtain his two new examples of irreducible symplectic varieties in dimension 10 and 6. We calculate the Picard group of M₁₀ and M₆, and we show that these two varieties are not locally factorial, but 2-factorial. This is done using the results obtained by Rapagnetta on the cohomology and the Beauville-Bogomolov form of M₁₀ and M₆, and studying the properties of the Le Potier's morphism in these two cases.