Mathematical prerequisites for stable-family extension

Written by GPT-6.1 Sol (OpenAI), Ultra, October 2026. Self-checked by the writing AI. Public domain (CC0 1.0).

The stable-family proof uses numerical curve models, surface resolution, curve stacks and Picard level structures. The numerical reading proves the weighted-graph classification, genus inequalities, long-chain estimate and torsion bound. Numerical identities give the corrected weighted diagrams and determinants. Regular surface models give the surface argument, including the normalized dominant component and the exceptional-curve contraction.

The geometric parameter construction uses the fixed-polynomial projective Hilbert/Quot theorem in Hilbert and Quot schemes, and the flat coherent-sheaf Hom/Isom and Picard constructions in Moduli stacks are algebraic. The curve-stack construction, infinitesimal automorphisms and smoothing are in The stack of curves. The stable-family reading constructs the genus piece it needs; it does not use a merely stated general curve-stack algebraicity theorem as its proof.

The dense-locus valuative criterion is Stable reduction and properness, Theorem 6.0 and Sections 6.1.1–6.1.7. Those sections construct the stack Chow support and keep the prescribed generic comparison arrow. Proposition 6.1 proves separatedness by extending marked stable-model isomorphisms. The ordinary scheme Zariski Main and prescribed-open approximation prerequisites recorded in that reading retain their stated scope and current programme status.

The torsion-visibility argument uses the curve Picard scheme and the étale kernel of invertible multiplication on a Jacobian. Abelian varieties supplies the degree, étaleness and Tate-module calculations. The stable-family reading proves the finite separable descent of the line-bundle classes, the reduced-curve torsion rank, its equality case, specialization, level rigidity and the descent across resolved nodes. Complete-local finite-étale existence and Stein splitting use Proper fundamental groups, Theorems 2.1 and 5.1, followed by faithful flat descent.

Ordinary foundations remain in force: proper coherent finiteness, formal functions, curve duality and Riemann–Roch; Cohen structure and local duality for complete surface rings; excellence and finite normalization; finite-presentation descent, proper quasi-finite finiteness and étale descent. The arguments are in the excellent Noetherian variety, trait and atlas settings stated in the proof readings. No arbitrary nonexcellent normalization assertion is used.

For the valuative properness argument the dominating trait has algebraically closed residue field. The unrestricted imperfect-residue-field form of the numerical semistability claim is not asserted. The level space is finite étale on the compact-type open; the final scheme cover uses the proved separable weak Chow construction and retains the universal family by pullback.