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Intro; Preface; Acknowledgements; References; Contents; Notations and Conventions; References; 1 Zero-Dimensional Schemes for Singularities; 1.1 Cluster and Zero-Dimensional Schemes; 1.1.1 Constellations and Cluster; 1.1.2 Cluster Schemes and Equisingularity; 1.1.3 The Hilbert Scheme of Cluster Schemes; 1.1.4 Zero-Dimensional Schemes for Analytic Types; 1.2 Non-classical Singularity Invariants; 1.2.1 Determinacy Bounds; 1.2.2 New Topological Invariants; 1.2.3 New Analytic Invariants; 1.3 Historical Notes and References; References; 2 Global Deformation Theory; 2.1 Classical Global Theorems
2.1.1 Divisors and Linear Systems2.1.2 Bézout's Theorem; 2.1.3 Bertini's and Noether's Theorems; 2.1.4 Polar and Dual Curves; 2.2 Equisingular Families of Singular Algebraic Varieties; 2.2.1 Families with Imposed Conditions on Singularities; 2.2.2 Hilbert Schemes of Singular Hypersurfaces; 2.2.3 T-Smooth Families of Isolated Hypersurface Singularities; 2.3 Construction via Deformation; 2.3.1 General Idea of the Patchworking Construction; 2.3.2 Polytopes and calS-transversality; 2.3.3 Gluing Singular Hypersurfaces; 2.3.4 SQH and NND Isolated Hypersurface Singularities
2.3.5 Lower Deformations of Hypersurface Singularities2.3.6 Cohomology Vanishing Conditions; 2.4 Appendix: The Patchworking Construction; 2.4.1 Elements of Toric Geometry; 2.4.2 Viro's Theorem for Hypersurfaces; 2.4.3 Viro's Method in Real Algebraic Geometry; 2.4.4 Viro's Theorem for Complete Intersections; 2.4.5 Other Examples of Patchworking; 2.5 Historical Notes and References; References; 3 H1-Vanishing Theorems; 3.1 Riemann-Roch Type H1-Vanishing; 3.2 Applications of Kodaira Vanishing; 3.3 Reider-Bogomolov Theory; 3.4 The Horace Method; 3.4.1 The Basic Horace Method
3.4.2 Applications of the Basic Horace Method3.4.3 Variations of the Horace Method; 3.5 The Castelnuovo Function; 3.6 H1-Vanishing for Generic Zero-Dimensional Schemes; 3.7 Historical Notes and References; References; 4 Equisingular Families of Curves; 4.1 Overview of New Results and Methods; 4.1.1 Nonemptiness; 4.1.2 T-Smoothness and Deformation Completeness; 4.1.3 Irreducibility; 4.1.4 Comments on the Methods; 4.2 Formulation of Problems, Discussion of Results, Examples; 4.2.1 Statement of the Problem; 4.2.2 Curves with Nodes and Cusps: from Severi to Harris
4.2.3 Examples of Obstructed and Reducible ESF4.3 T-Smoothness; 4.3.1 Linear Conditions; 4.3.2 Quadratic Conditions; 4.3.3 Obstructed Equisingular Families; 4.3.4 ESF of Hypersurfaces in mathbbPn; 4.4 Independence of Simultaneous Deformations; 4.4.1 Joint Versal Deformations; 4.4.2 1-Parametric Deformations; 4.5 Existence; 4.5.1 Conditions for the Patchworking Construction; 4.5.2 Plane Curves with Nodes and Cusps; 4.5.3 Curves and Hypersurfaces with Simple Singularities; 4.5.4 Hypersurfaces with Arbitrary Singularities; 4.5.5 Plane Curves with Arbitrary Singularities
2.1.1 Divisors and Linear Systems2.1.2 Bézout's Theorem; 2.1.3 Bertini's and Noether's Theorems; 2.1.4 Polar and Dual Curves; 2.2 Equisingular Families of Singular Algebraic Varieties; 2.2.1 Families with Imposed Conditions on Singularities; 2.2.2 Hilbert Schemes of Singular Hypersurfaces; 2.2.3 T-Smooth Families of Isolated Hypersurface Singularities; 2.3 Construction via Deformation; 2.3.1 General Idea of the Patchworking Construction; 2.3.2 Polytopes and calS-transversality; 2.3.3 Gluing Singular Hypersurfaces; 2.3.4 SQH and NND Isolated Hypersurface Singularities
2.3.5 Lower Deformations of Hypersurface Singularities2.3.6 Cohomology Vanishing Conditions; 2.4 Appendix: The Patchworking Construction; 2.4.1 Elements of Toric Geometry; 2.4.2 Viro's Theorem for Hypersurfaces; 2.4.3 Viro's Method in Real Algebraic Geometry; 2.4.4 Viro's Theorem for Complete Intersections; 2.4.5 Other Examples of Patchworking; 2.5 Historical Notes and References; References; 3 H1-Vanishing Theorems; 3.1 Riemann-Roch Type H1-Vanishing; 3.2 Applications of Kodaira Vanishing; 3.3 Reider-Bogomolov Theory; 3.4 The Horace Method; 3.4.1 The Basic Horace Method
3.4.2 Applications of the Basic Horace Method3.4.3 Variations of the Horace Method; 3.5 The Castelnuovo Function; 3.6 H1-Vanishing for Generic Zero-Dimensional Schemes; 3.7 Historical Notes and References; References; 4 Equisingular Families of Curves; 4.1 Overview of New Results and Methods; 4.1.1 Nonemptiness; 4.1.2 T-Smoothness and Deformation Completeness; 4.1.3 Irreducibility; 4.1.4 Comments on the Methods; 4.2 Formulation of Problems, Discussion of Results, Examples; 4.2.1 Statement of the Problem; 4.2.2 Curves with Nodes and Cusps: from Severi to Harris
4.2.3 Examples of Obstructed and Reducible ESF4.3 T-Smoothness; 4.3.1 Linear Conditions; 4.3.2 Quadratic Conditions; 4.3.3 Obstructed Equisingular Families; 4.3.4 ESF of Hypersurfaces in mathbbPn; 4.4 Independence of Simultaneous Deformations; 4.4.1 Joint Versal Deformations; 4.4.2 1-Parametric Deformations; 4.5 Existence; 4.5.1 Conditions for the Patchworking Construction; 4.5.2 Plane Curves with Nodes and Cusps; 4.5.3 Curves and Hypersurfaces with Simple Singularities; 4.5.4 Hypersurfaces with Arbitrary Singularities; 4.5.5 Plane Curves with Arbitrary Singularities