Moduli of Double EPW-Sextics

Moduli of Double EPW-Sextics Synopsis

The author studies the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of $\bigwedge^3{\mathbb C}^6$ modulo the natural action of $\mathrm{SL}_6$, call it $\mathfrak{M}$. This is a compactification of the moduli space of smooth double EPW-sextics and hence birational to the moduli space of HK $4$-folds of Type $K3^{[2]}$ polarized by a divisor of square $2$ for the Beauville-Bogomolov quadratic form. The author will determine the stable points. His work bears a strong analogy with the work of Voisin, Laza and Looijenga on moduli and periods of cubic $4$-folds.