Moduli of Double EPW-Sextics.
- 1st ed.
- 1 online resource (188 pages)
- Memoirs of the American Mathematical Society Series ; v.240 .
- Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Chapter 0. Introduction -- Chapter 1. Preliminaries -- 1.1. EPW-sextics and their double covers -- 1.2. Double EPW-sextics modulo isomorphisms -- 1.3. The GIT quotient -- 1.4. Moduli of plane sextics -- Chapter 2. One-parameter subgroups and stability -- 2.1. Outline of the section -- 2.2. (Semi)stability and flags -- 2.3. The Cone Decomposition Algorithm -- 2.4. The standard non-stable strata -- 2.4.1. The definitions -- 2.4.2. Geometric significance of certain strata -- 2.5. The stable locus -- 2.6. The GIT-boundary -- Chapter 3. Plane sextics and stability of lagrangians -- 3.1. The main result of the chapter -- 3.2. Plane sextics -- 3.3. Non-stable strata and plane sextics, I -- 3.4. Non-stable strata and plane sextics, II -- Chapter 4. Lagrangians with large stabilizers -- 4.1. Main results -- 4.2. A result of Luna -- 4.3. Lagrangians stabilized by a maximal torus -- 4.4. Lagrangians stabilized by \PGL(4) or \PSO(4) -- 4.5. Lagrangians stabilized by \PGL(3) -- Chapter 5. Description of the GIT-boundary -- 5.1. Main results -- 5.2. A GIT set-up for each standard non-stable stratum -- 5.2.1. Set-up -- 5.2.2. The Hilbert-Mumford numerical function -- 5.3. Summary of results of Chapters 6 and 7 -- 5.4. Proof of Theorem 5.1.1 assuming the results of Chapters 6 and 7 -- 5.4.1. Dimensions -- 5.4.2. No inclusion relations -- Chapter 6. Boundary components meeting ℑ in a subset of _∪} -- 6.1. \gB_ -- 6.1.1. First results -- 6.1.2. Properly semistable points of ^_ -- 6.1.3. Semistable lagrangians with dimΘ_≥2 or _=\PP( ). -- 6.1.4. Analysis of Θ_ and _ -- 6.1.5. Wrapping it up -- 6.2. \gB_ -- 6.2.1. The GIT analysis -- 6.2.2. Analysis of Θ_ and _ -- 6.2.3. Wrapping it up -- 6.3. \gB_ -- 6.3.1. Quadrics associated to ∈ ^_ -- 6.3.2. The GIT analysis. 6.3.3. Analysis of Θ_ and _ -- 6.3.4. Wrapping it up -- 6.4. \gB_ -- 6.4.1. The GIT analysis -- 6.4.2. Analysis of Θ_ and _ -- 6.4.3. Wrapping it up -- 6.5. \gB_₁} -- 6.5.1. The GIT analysis -- 6.5.2. Analysis of Θ_ and _ -- 6.5.3. Wrapping it up -- 6.6. \gB_ -- 6.6.1. The GIT analysis -- 6.6.2. Analysis of Θ_ and _ -- 6.6.3. Wrapping it up -- Chapter 7. The remaining boundary components -- 7.1. \gB_ -- 7.2. \gB_∩\gI -- 7.2.1. Set-up and statement of the main results -- 7.2.2. The 3-fold swept out by the projective planes parametrized by ₊( ) -- 7.2.3. Explicit description of \WW^_. -- 7.2.4. \gX_ is irreducible of dimension 3 -- 7.2.5. Points of \gB_∩\gI are represented by lagrangians in \WW^_ -- 7.2.6. _ for ∈\XX^_ and spanned by ∈ ₀₁, ∈ ₂₃ and ∈ ₄₅ -- 7.2.7. Proof that \gB_∩\gI=\gX_ -- 7.3. \gX_ -- 7.4. \gX_∩\gI -- 7.4.1. Set-up and statement of the main results -- 7.4.2. Duality -- 7.4.3. Properties of \gX_ -- 7.4.4. Points of \gX_∩\gI are represented by lagrangians in \YY^_ -- 7.4.5. Proof that \gX_∩\gI=\gX_∪\gX_ -- Appendix A. Elementary auxiliary results -- A.1. Discriminant of quadratic forms -- A.2. Quadratic forms of corank 2 -- A.3. Pencils of degenerate linear maps -- Appendix B. Tables -- Bibliography -- Back Cover.
The author studies the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of \bigwedge^3^6 modulo the natural action of \mathrm_6, call it \mathfrak. 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^ 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.