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E-book: Moduli of Weighted Hyperplane Arrangements

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This book focuses on a large class of geometric objects in moduli theory and provides explicit computations to investigate their families. Concrete examples are developed that take advantage of the intricate interplay between Algebraic Geometry and Combinatorics. Compactifications of moduli spaces play a crucial role in Number Theory, String Theory, and Quantum Field Theory – to mention just a few. In particular, the notion of compactification of moduli spaces has been crucial for solving various open problems and long-standing conjectures. Further, the book reports on compactification techniques for moduli spaces in a large class where computations are possible, namely that of weighted stable hyperplane arrangements (shas).
Preface vii
Introduction 1(2)
1 Stable Pairs and Their Moduli
1.1 The curve case
3(2)
1.2 Minimal model program: main definitions and results
5(5)
1.3 Minimal model program and one-parameter degenerations
10(4)
1.4 Moduli of stable varieties
14(3)
1.5 Moduli of stable pairs (X, B) with B # 0
17(4)
1.6 Moduli of stable varieties and pairs: known cases
21(2)
2 Stable Toric Varieties
2.1 Projective toric varieties and polytopes
23(3)
2.2 Stable toric varieties and tilings
26(2)
2.3 Linear systems on toric and stable toric varieties
28(2)
2.4 Stable toric varieties over a projective variety V
30(2)
2.5 Stable toric pairs vs stable toric varieties over Pn--1
32(1)
2.6 Singularities of stable toric varieties
32(2)
2.7 One-parameter degenerations
34(4)
2.8 Toric varieties associated to hyperplane arrangements
38(5)
3 Matroids
3.1 Vector (or representable) matroids
43(2)
3.2 Abstract matroids
45(2)
3.3 Connected components of a matroid
47(2)
3.4 Matroids of rank 1
49(1)
3.5 Matroids of rank 2
49(1)
3.6 Matroids of rank 3
50(3)
3.7 Flats
53(1)
3.8 Restrictions and contractions
53(1)
3.9 Dual matroids
54(1)
3.10 Regular matroids and degenerations of abelian varieties
54(6)
4 Matroid Polytopes and Tilings
4.1 Base polytope and independent set polytope
60(2)
4.2 Facets and faces
62(1)
4.3 Matroid polytopes and log canonical singularities
63(1)
4.4 Cuts of polytopes and log canonical singularities
64(2)
4.5 Matroid tilings
66(1)
4.6 Rank-2 case
66(2)
4.7 Rank-3 case
68(4)
4.8 Tropical projective spaces and Dressians
72(1)
4.9 Dual matroid polytopes and dual tilings
73(1)
4.10 Mnev's universality theorem
74(1)
5 Weighted Stable Hyperplane Arrangements
5.1 GIT and VGIT
75(5)
5.2 Semi-log canonical singularities and GIT
80(1)
5.3 Weighted shas
81(2)
5.4 Moduli spaces of shas
83(1)
5.5 Geography of the moduli spaces of shas
84(1)
5.6 Shas of dimension 1
85(1)
5.7 Shas of dimension 2
86(7)
6 Abelian Galois Covers
6.1 The yoga of cyclic and abelian Galois covers
93(3)
6.2 Special K3 surfaces
96(3)
6.3 Numerical Campedelli surfaces
99(1)
6.4 Kulikov surfaces
99(2)
Bibliography 101
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