Explicit study of structures of irreducible symplectic manifolds

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O’Grady constructed two sporadic examples of compact irreducible symplectic Kahler manifolds using moduli spaces of rank 2 semi-stable coherent sheaves on a K3 surface or an Abelian surface. I studied the locus of non-locally free sheaves in these moduli spaces and clarified some birational-geometric structures on the O’Grady’s examples. In the course of study, I investigated a method to calculate explicitly some invariant rings associated to the moduli problem using classical invariant theory, with the aid of computational algebra.