Foundations and dependency boundaries
The stable-family proof is reconstructed in the main reader and the regular surface-model bridge. Self-checked by the writing AI.
The following ordinary foundations remain explicit:
| Scope | Exact access and boundary |
|---|---|
| Proper coherent finiteness, base change, formal functions, cohomological dimension of curves | Full native cohomology and direct dependency chapters are indexed in the native manifest; their ordinary results are retained, and not every transitive foundation is proved here. |
| Fixed-polynomial projective Hilbert/Quot and sheaf Hom/Isom | The exact AG-HP Hilbert theorem and AG-AS flat coherent-sheaf/Hom-Isom proofs are bundled. The general curve-stack theorem in AG-AS §8 is stated and is not the construction used here. |
| Excellence, finite normalization, compatibility with étale base change and normal completion | Retained in the excellent Noetherian varieties/traits/stack-atlas scope. Full direct native algebra and More-on-Algebra sources are included; arbitrary nonexcellent normalization is not claimed. |
| Cohen complete regular subrings | Used in the complete local surface induction. This is an explicit ordinary foundation, not an assertion newly proved by the surface bridge. |
| Grothendieck duality, Matlis/local duality, coherent proper/formal-function finiteness | Used by the actual surface vanishing and boundedness proofs. The full resolution chapter is bundled, including those proof bodies, with the direct native dependency chapters. The entire transitive duality corpus is not proved here. |
| Domination of a modification by a blowup; finiteness after blowing up | The complete original flat.tex includes flat-lemma-dominate-modification-by-blowup and flat-lemma-finite-after-blowing-up, with pinned labels and full proof bodies. |
| Riemann–Roch, Riemann–Hurwitz, norm/pullback, smooth-curve Picard variety | Ordinary curve foundations are retained with exact native access. The smooth-component rigidity argument is independent norm/Riemann–Hurwitz exposition, not copied protected expression from the primary PDFs. |
| Proper/quasi-finite finiteness; separated étale descent; finite-unramified algebra | Used in level inertia, normalization, the node sections and Stein splitting. These are ordinary scheme/space foundations in the stated finite-presentation/Noetherian scope. |
| Regular-local factoriality and nodal resolution | Used to extend trait line bundles; nodal blowups are the exact AG-AS construction. The full regular-surface resolution and contraction proofs are included, not replaced by a bibliography title. |
| Complete-local existence and Stein idempotent lifting | The exact AG-DFG complete-local Theorem2.1 and Theorem5.1 are used with faithful flat descent. Its separately stated arbitrary henselian equivalence is not used as a proved input. |
Only NS.11a is imported from the existing AG-GS semistable supplement. Its following unrestricted imperfect-residue NS.11 is not proved here. Native 0CEE is used solely after a dominating trait with algebraically closed residue field; no rational-articulation/leaf repair is used.
The level-space proof claims finite étaleness over the compact-type open. The stronger everywhere finite projective fine pointed-moduli cover remains unclaimed. The scheme alteration is obtained by the exact separable weak Chow construction and retains the universal family by pullback.