Moduli of double EPW-sextics

Abstract

We will study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of 3 C 6 modulo the natural action of SL6, call it 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. We will determine the stable points. Our work bears a strong analogy with the work of Voisin, Laza and Looijenga on moduli and periods of cubic 4-folds. We will prove a result which is analogous to a theorem of Laza asserting that cubic 4-folds with simple singularities are stable. We will also describe the irreducible components of the GIT boundary of M. Our final goal (not achieved in this work) is to understand completely the period map from M to the Baily-Borel compactification of the relevant period domain modulo an arithmetic group. We will analyze the locus in the GIT-boundary of M where the period map is not regular. Our results suggest that M is isomorphic to Looijenga's compactification associated to 3 specific hyperplanes in the period domain.

Periods of double EPW-sextics

“…Suppose that W ∈ Θ A : there is a natural determinantal subscheme C W,A ⊂ P(W ), see Subsect. 3.2 of [23], with the property that…”

“…(Notice that if A ∈ (LG( 3 V ) \ Σ) then A is a stable point (Cor. 2.5.1 of [23]) and hence [A] ∈ M ADE because Θ A is empty.) In the present paper we will give a preliminary result in that direction.…”

“…be semistable with orbit closed in LG( 3 V ) ss and representing x. Results of [23] and [21] give that dim Θ A ≤ 1; this result combined with the regularity of (0.0.14) at all (W, A) with W ∈ Θ A gives that dim(p −1 (A) ∩ Σ) ≤ 1: that contradicts (0.0.12) and hence proves that p is regular at x (it proves also the last clause in the statement of Theorem 0.2). In Section 4 we will prove Theorem 0.3.…”

On IHS fourfolds with $b_2 = 23$

“…It follows that the corresponding EPW sextic is a double cubic. Since dim(P(A) ∩ F v ) = 5, it follows from [O4,Prop. 3.1.2] and [O4,Claim 3.2.2] that v is a point of multiplicity 6 on C U,A for U ∈ D 5 .…”

“…Since dim(P(A) ∩ F v ) = 5, it follows from [O4,Prop. 3.1.2] and [O4,Claim 3.2.2] that v is a point of multiplicity 6 on C U,A for U ∈ D 5 . Thus C U,A is a sum of multiple lines passing through v (if it is the whole plane we obtain a contradiction).…”