Periods of Double EPW-sextics

TL;DR

This paper investigates the indeterminacy of the period map for double EPW-sextics, focusing on the geometric conditions of degeneracy subschemes associated with lagrangian subspaces.

Contribution

It characterizes the indeterminacy locus in terms of degeneracy subschemes and establishes conditions for the regularity of the period map based on GIT stability.

Findings

The indeterminacy locus is contained in lagrangians with decomposable vectors.

Degeneracy subschemes are either entire planes or sextic curves.

The period map is regular when degeneracy sextics are GIT-semistable and avoid triple conics.

Abstract

We study the indeterminacy locus of the period map for double EPW-sextics. We recall that double EPW-sextics are parametrized by lagrangian subspaces of the third wedge-product of a 6-dimensional complex vector-space. The indeterminacy locus is contained in the set of lagrangians containing a decomposable vector. The projectivization of the 3-dimensional support of such a decomposable vector contains a degeneracy subscheme which is either all of the plane or a sextic curve. We show that the period map is regular on any lagrangian A such that for all decomposables in A the corresponding degeneracy subscheme is a GIT-semistable sextic curve whose closure (in the semistable locus) does not contain a triple conic.