Mathematics – Algebraic Geometry
Scientific paper
2011-11-06
Mathematics
Algebraic Geometry
Third in a series of papers on moduli and periods of double EPW-sextics
Scientific paper
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: there are strong analogies with the moduli space of cubic 4-folds. We determine the stable points, the irreducible components of the GIT boundary and their dimensions. Our final goal is to understand the period map from M to the Baily-Borel compactification of the relevant period domain modulo an arithmetic group. With this motivation in mind we prove a result which is analogous to a theorem of Laza on cubic 4-folds with simple singularities namely stability of lagrangians whose associated period point lands in the interior of the Baily-Borel compactification. We also analyze the locus in the GIT-boundary of M where the period map is not regular (presumably the indeterminacy locus is contained in the GIT boundary). It turns out to have two irreducible components, one of dimension 3, the other of dimension 1. Our result strongly suggests that M is isomorphic to Looijenga's compactification associated to 3 specific hyperplanes in the period domain.
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