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Intro; Foreword; Preface; Contents; Stratifying Quotient Stacks and Moduli Stacks; 1 Introduction; 2 Classical and Non-reductive GIT; 2.1 Classical GIT for Reductive Groups; 2.1.1 The HKKN Stratification; 2.1.2 Partial Desingularisations of Reductive GIT Quotients; 2.2 GIT for Non-reductive Groups; 2.3 GIT for Linear Algebraic Groups with Internally Graded Unipotent Radicals; 3 Stratifying Quotient Stacks; 4 Stacks Filtered by Quotient Stacks; 5 Applications; 5.1 Moduli of Hypersurfaces in a Complete Toric Variety; 5.2 Moduli of Sheaves Over a Projective Scheme; 5.3 Moduli of Unstable Curves
3.2 Tangent Space3.3 Elementary Modifications; 3.4 Comparing Components; 3.4.1 Separating Two Loci; 3.4.2 Separating Multiple Loci; 4 Brill-Noether Theorems; 4.1 Rank 1 Sheaves; 4.2 The Projective Plane; 4.3 Hirzebruch Surfaces; 4.4 Del Pezzo Surfaces and More General Rational Surfaces; 4.5 Other Surfaces; References; Geometric Invariant Theory of Syzygies, with Applications to Moduli Spaces; 1 Introduction; 1.1 Notation and Conventions; 2 Syzygy Points; 3 Families of Polarized Varieties; 3.1 Summation Formulae; 4 K3 Surfaces; 4.1 Effective Divisors on the Moduli Space of K3 Surfaces
5 Canonical Curves5.1 Hassett-Keel Program for Mg and Syzygies; 6 GIT for Syzygies of Canonical Genus Six Curves; 6.1 Clifford Index 1; 6.2 Clifford Index 2; 6.2.1 Bielliptic Curves and Elliptic Ribbons; 6.2.2 Singular Degree 5 del Pezzo; 6.3 The Hassett-Keel Program for M6; 6.3.1 Future Directions; References; The Topology of Ag and Its Compactifications; 1 Introduction; 2 The Complex Analytic Approach; 3 The Tautological Ring of Ag; 4 Compactifications and the Proportionality Principle; 5 L2 Cohomology and Zucker's Conjecture; 6 Computations in Small Genus; 7 Stabilization Results
3.2 Tangent Space3.3 Elementary Modifications; 3.4 Comparing Components; 3.4.1 Separating Two Loci; 3.4.2 Separating Multiple Loci; 4 Brill-Noether Theorems; 4.1 Rank 1 Sheaves; 4.2 The Projective Plane; 4.3 Hirzebruch Surfaces; 4.4 Del Pezzo Surfaces and More General Rational Surfaces; 4.5 Other Surfaces; References; Geometric Invariant Theory of Syzygies, with Applications to Moduli Spaces; 1 Introduction; 1.1 Notation and Conventions; 2 Syzygy Points; 3 Families of Polarized Varieties; 3.1 Summation Formulae; 4 K3 Surfaces; 4.1 Effective Divisors on the Moduli Space of K3 Surfaces
5 Canonical Curves5.1 Hassett-Keel Program for Mg and Syzygies; 6 GIT for Syzygies of Canonical Genus Six Curves; 6.1 Clifford Index 1; 6.2 Clifford Index 2; 6.2.1 Bielliptic Curves and Elliptic Ribbons; 6.2.2 Singular Degree 5 del Pezzo; 6.3 The Hassett-Keel Program for M6; 6.3.1 Future Directions; References; The Topology of Ag and Its Compactifications; 1 Introduction; 2 The Complex Analytic Approach; 3 The Tautological Ring of Ag; 4 Compactifications and the Proportionality Principle; 5 L2 Cohomology and Zucker's Conjecture; 6 Computations in Small Genus; 7 Stabilization Results