By Kieran G. O'grady

The writer reports the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of $\bigwedge^3{\mathbb C}^6$ modulo the usual motion of $\mathrm{SL}_6$, name it $\mathfrak{M}$. it is a compactification of the moduli area of delicate double EPW-sextics and as a result birational to the moduli area of HK $4$-folds of variety $K3^{[2]}$ polarized by way of a divisor of sq. $2$ for the Beauville-Bogomolov quadratic shape. the writer will verify the good issues. His paintings bears a powerful analogy with the paintings of Voisin, Laza and Looijenga on moduli and classes of cubic $4$-folds.

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**Moduli of Double Epw-sextics (Memoirs of the American Mathematical Society)**

The writer reviews the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of $\bigwedge^3{\mathbb C}^6$ modulo the average motion of $\mathrm{SL}_6$, name it $\mathfrak{M}$. this can be a compactification of the moduli house of tender double EPW-sextics and as a result birational to the moduli house of HK $4$-folds of style $K3^{[2]}$ polarized by way of a divisor of sq. $2$ for the Beauville-Bogomolov quadratic shape.

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**Additional info for Moduli of Double Epw-sextics (Memoirs of the American Mathematical Society)**

**Example text**

Let U ⊂ V be complementary to W . 12) ( 2 W) ∧ U ⊕ W ∧ ( ∼ U ) −→ EW . 13) EW = ([v0 ] ∧ W0 ∧ U ⊕ [v0 ] ∧ ( 2 U )) ⊕ (( 2 W0 ) ∧ U ⊕ W0 ∧ ( U )). ) Given the above decomposition the scheme CW,A ∩ (P(W ) \ P(W0 )) is described as the degeneracy locus of a family of quadratic forms. One identiﬁes the family of quadratic forms with {(q A + q w )}w∈W0 and the claim follows. Licensed to Tulane Univ. 78. org/publications/ebooks/terms 42 3. 3. 5). 14) ZW0 ,A := V (q A ) ∩ Gr(2, V0 )W0 ⊂ P( 2 V0 / W0 ).

A ∈ Σ∞ ) or A ∈ Σ[2]. By deﬁnition we may assume that A ∈ BFX for X one of A, A∨ , . . , F2 , or A ∈ XFN3 , where F is the basis {v0 , . . , v5 } of V . 2. It remains to consider A ∈ (BFC1 ∪ BFE1 ∪ BFE ∨ ∪ BFF2 ∪ XFN3 ). 1) one easily checks the following: 1 3 2 If A ∈ (BFC1 ∪ BFE1 ∪ BFE ∨ ) then V02 ⊂ A and dim(A ∩ ( V02 ∧ V )) ≥ 3, if 1 A ∈ (BFF2 ∪ XFN3 ) there exists a 3-dimensional subspace W ⊂ V03 containing V01 such that 3 W ⊂ A and dim(A ∩ ( 2 W ∧ V )) ≥ 3. 5 for details. 6. The GIT-boundary Let M ⊂ M be the (open) subset parametrizing PGL(V )-orbits of stable points; the GIT-boundary of M is ∂M := (M \ Mst ).

3), with weights in decreasing order: e0 > e1 > . . > es . 3) dX = (d0 , d1 , . . , d[(s−1)/2] ). 1) that contains BFX (or 3 XN3 ) . Let SFX ⊂ LG( V ) be the set of A which are λX -split of type dX . 4 gives the following: Licensed to Tulane Univ. 78. org/publications/ebooks/terms 30 2. 1. Every point of BX is represented by a point of SFX and every point of XN3 is represented by a point of SFN3 . Next let F be the basis of V obtained by reading the vectors in F in reverse order: F := {v5 , v4 , v3 , v2 , v1 , v0 }.