Resolution of singularities in characteristic zero

Functorial resolution of singularities for schemes of finite type over a field of characteristic zero: smooth blow-ups, derivative ideals, maximal contact, order reduction for ideals and marked ideals, principalization, resolution with a simple normal crossings exceptional divisor, smooth compactification and elimination of indeterminacy.

Ten lessons with proofs, examples and exercises with solutions. Programme text by Claude Opus 5.5 (Anthropic), with mathematical editing and reader preparation by GPT-6.1 Sol (OpenAI). Original programme text: public domain (CC0).

1. Smooth blow-ups and transforms of ideals
2. Blow-up sequences and the main theorems
3. Derivative ideals under blowing up
4. Hypersurfaces of maximal contact
5. Logarithmic derivatives and going up
6. Uniqueness of maximal contact
7. Tuning of ideals
8. Order reduction for ideals
9. Order reduction for marked ideals
10. Principalization and resolution

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Regular local rings (Commutative algebra for geometry)
• Kähler differentials (Commutative algebra for geometry)
• Smooth algebras over a field and the Jacobian criterion (Commutative algebra for geometry)
• Completion (Commutative algebra for geometry)
• Coefficient rings and the Cohen structure theorem (Commutative algebra for geometry)
• Smooth morphisms · Flat, smooth and étale morphisms
• Poincaré duality for smooth varieties (Sites, topoi and étale cohomology)
• Facts on blowing up, strict transforms, Nagata compactification and smooth and étale morphisms are cited by tag from the Stacks project, through the AI-integrated edition of the Stacks project.