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. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

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
• Kähler differentials
• Smooth algebras over a field and the Jacobian criterion
• Completion
• Coefficient rings and the Cohen structure theorem
• Smooth morphisms · Flat, smooth and étale morphisms
• Poincaré duality for smooth varieties
• 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.