Non-locally-free locus of O'Grady's ten dimensional example (Q361847)

Let \(S\) be a projective \(K3\) surface with Picard number one. \textit{K. G. O'Grady}'s singular moduli space \(M\) is the moduli space of Gieseker-semistable sheaves on \(S\) of rank 2, \(c_1=0\), and \(c_2=4\) [J. Reine Angew. Math. 512, 49-117 (1999; Zbl 0928.14029)], whose desingularization gives a new example of an irreducible holomorphic symplectic manifold. By the universal property of \(M\) and a certain universal property of \(M^{DU}\), there is a projective morphism \(\varphi: M\to M^{DU}\) where \(M^{DU}\) is the Donaldson-Uhlenbeck compactification. The morphism \(\varphi\) turns out to be birational and the exceptional divisor \(B\) of \(\varphi\) is exactly the locus of non-locally free sheaves. The paper under review gives a completely explicit description of the fibers of the morphism \(\varphi\) restricted on \(B\). The author describes these fibers as GIT quotients and computes explicitly their homogeneous invariant rings.