Periods of double EPW-sextics

We study the period map for double EPW-sextics, which are varieties making up a locally versal family of polarized hyperkähler fourfolds. Double EPW-sextics are parametrized by Lagrangian subspaces of the third wedge-product of a 6-dimensional complex vector space. We prove that Lagrangians in the indeterminacy locus of the period map contain a decomposable 3 -vector enjoying very special properties. In addition we prove a result about periods of double EPW-sextics which are either smooth or have isolated singular points with projectivized tangent cone isomorphic to the incidence variety in P 2 × ( P 2 ) ∨ .