{
  "id": "a-singular-surface-with-free-tangent-sheaf",
  "title": "A singular surface with free tangent sheaf",
  "description": "A normal affine complex surface whose module of derivations is free of rank two although the surface is singular, after OpenAI's 2026 counterexample to the Lipman-Zariski conjecture. An explicit elliptic curve on the quadric with a line bundle of order four; a curve of genus 21 with fourth-order contact and a fourth root of its canonical bundle; traces, residues and Grothendieck's splitting theorem; the trace form and the residue data; two vector fields glued order by order near the curve; contraction to a singular point with free derivations; algebraization by finite determinacy with relations.",
  "units": [
    {
      "id": "derivations-and-the-lipman-zariski-problem",
      "title": "Derivations and the Lipman–Zariski problem",
      "reader": "derivations-and-the-lipman-zariski-problem.html",
      "source": "src/derivations-and-the-lipman-zariski-problem.md",
      "source_sha256": "9724a8fae34a03bfd85f1feba0bc7b516c38f13ef24245effa27e7cc31fe5fe7",
      "status": "author_self_checked"
    },
    {
      "id": "an-elliptic-curve-on-the-quadric",
      "title": "An elliptic curve on the quadric",
      "reader": "an-elliptic-curve-on-the-quadric.html",
      "source": "src/an-elliptic-curve-on-the-quadric.md",
      "source_sha256": "0c0b239b4c20f8580e717dea86b0c3b6d3c482a590bc330861034c16b2251276",
      "status": "author_self_checked"
    },
    {
      "id": "a-curve-of-genus-21-with-a-fourth-root-of-its-canonical-bundle",
      "title": "A curve of genus 21 with a fourth root of its canonical bundle",
      "reader": "a-curve-of-genus-21-with-a-fourth-root-of-its-canonical-bundle.html",
      "source": "src/a-curve-of-genus-21-with-a-fourth-root-of-its-canonical-bundle.md",
      "source_sha256": "ba86659ba7057980d56923b61b9aeba80b79a946a22c0e63e2885c6185ebc8f2",
      "status": "author_self_checked"
    },
    {
      "id": "traces-residues-and-vector-bundles-on-the-projective-line",
      "title": "Traces, residues and vector bundles on the projective line",
      "reader": "traces-residues-and-vector-bundles-on-the-projective-line.html",
      "source": "src/traces-residues-and-vector-bundles-on-the-projective-line.md",
      "source_sha256": "ebc57c58ada8d53039384c9f168e0301f04e0f87d6b5bbb90442175b6212e3be",
      "status": "author_self_checked"
    },
    {
      "id": "the-trace-form-and-the-residue-data",
      "title": "The trace form and the residue data",
      "reader": "the-trace-form-and-the-residue-data.html",
      "source": "src/the-trace-form-and-the-residue-data.md",
      "source_sha256": "cc497af8b5a404250fe4556222f2dbbe8288e05057df37697b8cc65bfb930b47",
      "status": "author_self_checked"
    },
    {
      "id": "gluing-vector-fields-the-first-two-orders",
      "title": "Gluing vector fields: the first two orders",
      "reader": "gluing-vector-fields-the-first-two-orders.html",
      "source": "src/gluing-vector-fields-the-first-two-orders.md",
      "source_sha256": "8192374544105dfbfcc6fdae880430705b380c9f425a9bfb59df70427c4531a9",
      "status": "author_self_checked"
    },
    {
      "id": "gluing-vector-fields-higher-orders-and-the-determinant",
      "title": "Gluing vector fields: higher orders and the determinant",
      "reader": "gluing-vector-fields-higher-orders-and-the-determinant.html",
      "source": "src/gluing-vector-fields-higher-orders-and-the-determinant.md",
      "source_sha256": "a6e3135f1904669e463472f9adebe99641790a778e4d11280c317bf6bf3796bc",
      "status": "author_self_checked"
    },
    {
      "id": "analytic-realization-and-contraction",
      "title": "Analytic realization and contraction",
      "reader": "analytic-realization-and-contraction.html",
      "source": "src/analytic-realization-and-contraction.md",
      "source_sha256": "8ffbfff3238eab58e5425f73584c1e8caf82bf1ba342a7c4ae9140bb57cb7476",
      "status": "author_self_checked"
    },
    {
      "id": "algebraization",
      "title": "Algebraization",
      "reader": "algebraization.html",
      "source": "src/algebraization.md",
      "source_sha256": "cdde30a2ba7e12cbbacf4cd3603dc1dd837afbb9faee25b114b830aed554a686",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "Kahler differentials, smooth algebras and the Jacobian criterion, regular local rings, Serre's normality criterion, completion, Artin-Rees and henselization (course Commutative algebra for geometry, AG-CA)",
    "cohomology of O(a,b) on P^1 x P^1, Serre duality and the canonical bundle, Serre's theorems and comparison theorems (course Cohomology of quasi-coherent sheaves, AG-QC)",
    "Riemann-Roch for smooth projective curves (course The abstract theory of ideals, NOE-ABS, lesson 6, Theorem 4.3)",
    "holomorphic functions of several variables, coherence, the analytic Nullstellensatz, Theorems A and B on Stein manifolds (course Complex analytic spaces and coherent sheaves)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}