Moduli of Double EPW-Sextics

188 Pages·2016·1.982 MB·English

by  Kieran G. O’Grady

MEMOIRS
of the
American Mathematical Society
Volume 240 • Number 1136 • Forthcoming
Moduli of double EPW-sextics
Kieran G. O’Grady
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
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MEMOIRS
of the
American Mathematical Society
Volume 240 • Number 1136 • Forthcoming
Moduli of double EPW-sextics
Kieran G. O’Grady
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
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http://www.loc.gov/publish/cip/.
DOI:http://dx.doi.org/10.1090/memo/1136
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Dedicato alla piccola Emma
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Contents
Chapter 0. Introduction 1
Chapter 1. Preliminaries 7
1.1. EPW-sextics and their double covers 7
1.2. Double EPW-sextics modulo isomorphisms 8
1.3. The GIT quotient 10
1.4. Moduli of plane sextics 11
Chapter 2. One-parameter subgroups and stability 13
2.1. Outline of the section 13
2.2. (Semi)stability and flags 14
2.3. The Cone Decomposition Algorithm 17
2.4. The standard non-stable strata 19
2.4.1. The definitions 19
2.4.2. Geometric significance of certain strata 22
2.5. The stable locus 25
2.6. The GIT-boundary 29
Chapter 3. Plane sextics and stability of lagrangians 31
3.1. The main result of the chapter 31
3.2. Plane sextics 31
3.3. Non-stable strata and plane sextics, I 35
3.4. Non-stable strata and plane sextics, II 40
Chapter 4. Lagrangians with large stabilizers 49
4.1. Main results 49
4.2. A result of Luna 49
4.3. Lagrangians stabilized by a maximal torus 50
4.4. Lagrangians stabilized by PGL(4) or PSO(4) 53
4.5. Lagrangians stabilized by PGL(3) 57
Chapter 5. Description of the GIT-boundary 59
5.1. Main results 59
5.2. A GIT set-up for each standard non-stable stratum 60
5.2.1. Set-up 60
5.2.2. The Hilbert-Mumford numerical function 62
5.3. Summary of results of Chapters 6 and 7 62
5.4. Proof of Theorem 5.1.1 assuming the results of Chapters 6 and 7 63
5.4.1. Dimensions 63
5.4.2. No inclusion relations 63
v
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vi CONTENTS
Chapter 6. Boundary components meeting I in a subset of XW ∪{x,x∨} 65
6.1. BC 65
1
6.1.1. First results 65
6.1.2. Properly semistable points of SF 66
C
6.1.3. Semistable lagrangians A with d1imΘ ≥2 or C =P(W). 69
A W,A
6.1.4. Analysis of Θ and C 70
A W,A
6.1.5. Wrapping it up 73
6.2. BA 74
6.2.1. The GIT analysis 74
6.2.2. Analysis of Θ and C 76
A W,A
6.2.3. Wrapping it up 77
6.3. BD 77
6.3.1. Quadrics associated to A∈SF 78
D
6.3.2. The GIT analysis 80
6.3.3. Analysis of Θ and C 83
A W,A
6.3.4. Wrapping it up 90
6.4. BE 90
1
6.4.1. The GIT analysis 91
6.4.2. Analysis of Θ and C 96
A W,A
6.4.3. Wrapping it up 99
6.5. BE∨ 100
1
6.5.1. The GIT analysis 101
6.5.2. Analysis of Θ and C 101
A W,A
6.5.3. Wrapping it up 103
6.6. BF 104
1
6.6.1. The GIT analysis 104
6.6.2. Analysis of Θ and C 105
A W,A
6.6.3. Wrapping it up 106
Chapter 7. The remaining boundary components 107
7.1. BF 107
2
7.2. BF ∩I 111
2
7.2.1. Set-up and statement of the main results 111
7.2.2. The3-foldsweptoutbytheprojectiveplanesparametrizedbyi (D)112
+
7.2.3. Explicit description of Wψ . 113
fix
7.2.4. XV is irreducible of dimension 3 114
7.2.5. Points of BF2 ∩I are represented by lagrangians in Wψfix 119
7.2.6. C for A∈Xψ and W spanned by α∈V , β ∈V and γ ∈V 123
W,A V 01 23 45
7.2.7. Proof that BF ∩I=XV 129
2
7.3. XN 129
3
7.4. XN ∩I 132
3
7.4.1. Set-up and statement of the main results 132
7.4.2. Duality 135
7.4.3. Properties of XZ 136
7.4.4. Points of XN3 ∩I are represented by lagrangians in Yψfix 141
7.4.5. Proof that XN ∩I=XW ∪XZ 144
3
Appendix A. Elementary auxiliary results 149
A.1. Discriminant of quadratic forms 149
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CONTENTS vii
A.2. Quadratic forms of corank 2 150
A.3. Pencils of degenerate linear maps 152
Appendix B. Tables 155
Bibliography 171
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Abstract
WewillstudytheG(cid:2)ITquotientofthesymplecticgrassmannianparametrizing
lagrangian subspaces of 3C6 modulo the natural action of SL , call it M. This is
6
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
analogoustoatheoremofLazaassertingthatcubic4-foldswithsimplesingularities
arestable. WewillalsodescribetheirreduciblecomponentsoftheGITboundaryof
M. Ourfinalgoal(notachievedinthiswork)istounderstandcompletelytheperiod
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.
ReceivedbytheeditorOctober6,2012and,inrevisedform,March7,2014.
ArticleelectronicallypublishedonOctober9,2015.
DOI:http://dx.doi.org/10.1090/memo/1136
2010 MathematicsSubjectClassification. Primary14J10,14L24,14C30.
Key wordsand phrases. GITquotient,periodmap,hyperk¨ahlervarieties.
TheauthorwassupportedbyPRIN2007.
The author is affiliatedwith Sapienza Universit`a di Roma,Dipartimento di MatematicaG.
Castelnuovo,Roma,Italy.
(cid:2)c2015 American Mathematical Society
ix
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