Double covers of quadratic degeneracy and Lagrangian intersection loci

Abstract

We explain a general construction of double covers of quadratic degeneracy loci and Lagrangian intersection loci based on reflexive sheaves. We relate the double covers of quadratic degeneracy loci to the Stein factorizations of the relative Hilbert schemes of linear spaces of the corresponding quadric fibrations. We give a criterion for these double covers to be nonsingular. These results are an extension of O’Grady’s construction of double covers of EPW sextics and provide an alternate construction of Iliev–Kapustka–Kapustka–Ranestad’s EPW cubes.