Moduli of double EPW-sextics
  We study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of \bigwedge^3{\mathbb C}^6 by the natural action of SL_6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics. We determine the stable points, the irreducible components of the GIT boundary and their dimensions. There are strong analogies with the moduli space of cubic 4-folds: we prove a result which is analogous to a theorem of Laza stating that cubic 4-folds with simple singularities are stable. Our final goal (not achieved in the present paper) is to understand the period map from M to the Baily-Borel compactification of the relevant period domain modulo an arithmetic group. With this goal in mind we analyze the locus in the GIT-boundary of M where the period map is not regular: we prove that it has two irreducible components, one of dimension 3, the other of dimension 1. Our results suggests that M is isomorphic to Looijenga's compactification associated to 3 specific hyperplanes in the period domain. 
Kieran G. O'Grady
math.AG