Double covers of quadratic degeneracy and Lagrangian intersection loci

TL;DR

This paper introduces a general method for constructing double covers of quadratic degeneracy and Lagrangian intersection loci using reflexive sheaves, relating them to Hilbert schemes and providing criteria for smoothness.

Contribution

It presents a new construction of double covers for quadratic degeneracy and Lagrangian loci, connecting them to Hilbert schemes and applying to EPW sextics and cubes.

Findings

Double covers relate to Stein factorizations of Hilbert schemes.

Criteria established for smoothness of the double covers.

Construction yields known EPW varieties like O'Grady's double EPW sextics.

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. As applications of these results, we show that the double covers of the EPW sextics obtained by our construction give O'Grady's double EPW sextics and that an analogous construction gives Iliev-Kapustka-Kapustka-Ranestad's EPW cubes.