Stable pointed-family extension

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

Start with Stable pointed-family extension, then use the proof guide to find the numerical and surface arguments.

The family-extension theorem concerns a projective, generically finite, generically separable alteration and the stable pointed family obtained by pullback. In positive characteristic its level is invertible and at least three. The numerical argument for properness uses a dominating trait with algebraically closed residue field. It does not assert the unrestricted imperfect-residue-field statement.

The companion readings are Regular surface models, Numerical completion, Numerical identities, Mathematical prerequisites, and Sources and licence.