Regular Solids and Isolated Singularities

Inhaltsverzeichnis

• I: Regular Solids and Finite Rotation Groups.
• §1. The Platonic Solids.
• §2. Convex Polytopes.
• §3. Regular Solids.
• §4. Enumeration and Realization of Regular Solids.
• §5. The Rotation Groups of the Platonic Solids.
• §6. Finite Subgroups of the Rotation Group SO(3).
• §7. Normal Subgroups.
• §8. Generators and Relations for the Finite Subgroups of SO(3).
• II: Finite Subgroups of SL(2, G) and Invariant Polynomials.
• §1. Finite Subgroups of SL(2, C).
• §2. Quaternions and Rotations.
• §3. Four-Dimensional Regular Solids.
• §4. The Orbit Spaces S3/G of the Finite Subgroups G of SU(2).
• §5. Generators and Relations for the Finite Subgroups of SL(2, C).
• §6. Invariant Divisors and Semi-Invariant Forms.
• §7. The Characters of the Invariant Divisors.
• §8. Generators and Relations for the Algebra of Invariant Polynomials.
• §9. The Affine Orbit Variety.
• III: Local Theory of Several Complex Variables.
• §1. Germs of Holomorphic Functions.
• §2. Germs of Analytic Sets.
• §3. Germs of Holomorphic Maps.
• §4. The Embedding Dimension.
• §5. The Preparation Theorem.
• §6. Finite Maps.
• §7. Finite and Strict Maps.
• §8. The Nullstellensatz.
• §9. The Dimension.
• §10. Annihilators.
• §11. Regular Sequences.
• §12. Complete Intersections.
• §13. Complex Spaces.
• IV: Quotient Singularities and Their Resolutions.
• §1. Germs of Invariant Holomorphic Functions.
• §2. Complex Orbit Spaces.
• §3. Quotient Singularities.
• §4. Modifications. Line Bundles.
• §5. Cyclic Quotient Singularities.
• §6. The Resolution of Cyclic Quotient Singularities.
• §7. The Cotangent Action.
• §8. Line Bundles with Singularities.
• §9. The Resolution of Non-Cyclic Quotient Singularities.
• §10. Plumbed Surfaces.
• §11. Intersection Numbers.
• §12. The Homology of Plumbed Surfaces.
• §13. TheFundamental Group of a Plumbed Surface Minus its Core.
• §14. Groups Determined by a Weighted Tree.
• §15. Topological Invariants.
• V: The Hierarchy of Simple Singularities.
• §1. Basic Concepts.
• §2. The Milnor Number.
• §3. Transformation Groups.
• §4. Families of Germs.
• §5. Finitely Determined Germs.
• §6. Unfoldings.
• §7. The Multiplicity.
• §8. Weighted Homogeneous Polynomials.
• §9. The Classification of Holomorphic Germs.
• §10. Three Series of Holomorphic Germs.
• §11. Simple Singularities.
• §12. Adjacency.
• §13. Conclusion and Outlook.
• References.