{
  "id": "resolution-of-singularities-in-characteristic-zero",
  "title": "Resolution of singularities in characteristic zero",
  "description": "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.",
  "units": [
    {
      "id": "smooth-blowups-and-transforms-of-ideals",
      "title": "Smooth blow-ups and transforms of ideals",
      "reader": "smooth-blowups-and-transforms-of-ideals.html",
      "source": "src/smooth-blowups-and-transforms-of-ideals.md",
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      "title": "Blow-up sequences and the main theorems",
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      "title": "Derivative ideals under blowing up",
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      "title": "Hypersurfaces of maximal contact",
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      "source": "src/hypersurfaces-of-maximal-contact.md",
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      "id": "logarithmic-derivatives-and-going-up",
      "title": "Logarithmic derivatives and going up",
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    {
      "id": "uniqueness-of-maximal-contact",
      "title": "Uniqueness of maximal contact",
      "reader": "uniqueness-of-maximal-contact.html",
      "source": "src/uniqueness-of-maximal-contact.md",
      "source_sha256": "42be589c8d1305cb031ca1efb5f1f74ae52f65ab1ee8644019287ded8c123173",
      "status": "available"
    },
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      "id": "tuning-of-ideals",
      "title": "Tuning of ideals",
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      "source": "src/tuning-of-ideals.md",
      "source_sha256": "3c0fa6b3ab4946e4bfd77c19fd661fcd0be61e2d0ef237434a582fd8a3cc1173",
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    },
    {
      "id": "order-reduction-for-ideals",
      "title": "Order reduction for ideals",
      "reader": "order-reduction-for-ideals.html",
      "source": "src/order-reduction-for-ideals.md",
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    },
    {
      "id": "order-reduction-for-marked-ideals",
      "title": "Order reduction for marked ideals",
      "reader": "order-reduction-for-marked-ideals.html",
      "source": "src/order-reduction-for-marked-ideals.md",
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    {
      "id": "principalization-and-resolution",
      "title": "Principalization and resolution",
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    }
  ],
  "prerequisites": [
    "AG-CA (regular local rings, Kahler differentials, completion, Cohen structure)",
    "AG-FSE lessons on smooth morphisms",
    "AI Integrated Stacks Project: blowing up, strict transforms, Nagata compactification, smooth and etale morphisms",
    "Etale cohomology lesson 18, section 2 (inverse-function criterion)"
  ],
  "status": "available",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0",
  "contributors": [
    {
      "name": "Claude Opus 5.5 (Anthropic)",
      "contribution": "Course text, mathematical exposition and exercises."
    },
    {
      "name": "GPT-6.1 Sol (OpenAI)",
      "contribution": "Mathematical editing and reader preparation."
    }
  ]
}