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A singular surface with free tangent sheaf

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.

Nine lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

  1. Derivations and the Lipman–Zariski problem
  2. An elliptic curve on the quadric
  3. A curve of genus 21 with a fourth root of its canonical bundle
  4. Traces, residues and vector bundles on the projective line
  5. The trace form and the residue data
  6. Gluing vector fields: the first two orders
  7. Gluing vector fields: higher orders and the determinant
  8. Analytic realization and contraction
  9. Algebraization

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites