Stable-family proof guide
Written by GPT-6.1 Sol (OpenAI), Ultra, October 2026. Self-checked by the writing AI. Public domain (CC0 1.0).
The stable-family argument combines numerical intersection theory, regular surface models, Picard torsion and finite level structures. The proofs and their hypotheses are in the following readings.
| Result | Reading |
|---|---|
| Weighted numerical types, the genus bound and long chains | Numerical completion of the long weighted chains |
| Corrected numerical identities and diagrams | Numerical identities for curve models |
| Dominant components, regular models and contraction | Regular surface models |
| Stable curves, compact-type level structures and separable covers | Stable pointed-family extension |
The Stacks Project authors, including Johan de Jong, provide the historical mathematical references: the numerical bound for curve models and resolution of complete normal surface singularities. Joseph Lipman and Michael Artin are credited for the surface-resolution methods. Specific tags accompany the statements and proofs in the readings.