Finite domination of surface modifications by a relative Hilbert scheme

Given: A normal Noetherian local domain $A$ of dimension two essentially of finite type over a field or complete equicharacteristic local ring, a finite normal local $A$-domain $B$, and a normal integral modification $Y\to \mathrm{Spec}B$.

def-axiom-of-choice. The Axiom of Choice (AC) is the following statement. > Every family of nonempty sets has a choice function > (def-choice-function). Written out: for every set $\mathcal{F}$ all of whose members are nonempty, there exists a function $g$ with domain $\mathcal{F}$ satisfying $g(S)\in S$ for all $S\in \mathcal{F}$. (The Axiom of Choice)

def-dependent-choice. Let $X$ be a set and let $R\subseteq X\times X$ be a binary relation on $X$. Call $R$ entire on $X$ when $\text{for every }x\in X\text{ there is }y\in X\text{ with }xRy\mathrm{.}$ The Axiom of Dependent Choice, written $\mathrm{D}\mathrm{C}$, is the following statement. (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

def-normal-surface-modification-and-normalized-point-blowup. Normal schemes. A locally Noetherian scheme is normal if every local ring ${\mathcal{O}}_{X,x}$ is an integrally closed domain (def-normal-noetherian-ring). This is a local condition on the local rings and is checked on an affine open cover; it does not require the global section ring to be a domain. The empty scheme is normal vacuously. (Normal scheme modifications and normalized point blowups)

lem-normalized-point-blowups-dominate-local-normal-surface-modifications. Assume AC and DC. Let $A$ be a normal two-dimensional Noetherian local domain essentially of finite type over a field or a complete equicharacteristic Noetherian local ring. Let $S$ be a normal integral modification of $\mathrm{Spec}A$, and let $Y\to S$ be an integral modification with normal $Y$. (Normalized point blowups dominate local normal surface modifications)

lem-surface-finite-type-normalization-finite. Assume AC and DC. Every integral finite-type algebra over a field or a complete equicharacteristic Noetherian local base has finite normalization, and so do its localizations. Integral schemes of finite type over these bases consequently have finite scheme normalization. (Surface finite type normalization finite)

lem-finite-over-projective-noetherian-affine-base-is-projective. Assume AC. Let $R$ be Noetherian, let $X$ be projective over $R$, and let $Y\to X$ be finite. Then $Y\to X$ admits a closed immersion into one relative projective space over $X$, and $Y$ is projective over $R$. Finite compositions of projective morphisms between such schemes are projective. (Finite schemes over projective schemes are projective over a Noetherian affine base)

thm-hilbert-scheme-represents-projective-flat-families. Assume AC and DC. Let $S$ be any locally Noetherian scheme, possibly non-quasi-compact, let $X\to S$ be projective of finite presentation in the convention of def-projective-morphism-coherent-bundle-convention, and let $L$ be relatively ample. (Projective Hilbert schemes represent all flat finitely presented families)

def-hilbert-functor-of-flat-projective-subschemes. Work with AC and DC. Fix a locally Noetherian scheme $S$, possibly non-quasi-compact, a projective morphism of finite presentation $X\to S$ in the convention of def-projective-morphism-coherent-bundle-convention, and a relatively ample invertible sheaf $L$ on $X$. The test category is all $S$-schemes, with arbitrary $S$-morphisms. Set $X_T=X{\times }_{S}T$. (Hilbert functor of flat finitely presented projective families)

thm-hilbert-polynomial-degree-support-dimension. Assume the Axiom of Choice, inherited from the Hilbert-polynomial, hyperplane, global-generation and base-change suppliers cited below (The Axiom of Choice). (Degree of the coherent Hilbert polynomial)

thm-proper-quasi-finite-is-finite. Assume the Axiom of Choice. Every proper quasi-finite morphism of schemes $f:X\to S$ is finite (def-proper-morphism, def-quasi-finite-morphism-schemes, def-finite-morphism-schemes). No Noetherian or nonemptiness hypothesis is imposed, and the assertion is local on the base. (A proper quasi-finite morphism is finite)

cor-finite-flat-noetherian-modules-are-projective. Let $R$ be a Noetherian commutative ring and let $M$ be a finite flat $R$-module. Then $M$ is finite projective. (A finite flat module over a Noetherian ring is finite projective)

lem-proper-source-to-separated-target-proper. Assume the Axiom of Choice. Let $f:X\to S$ be proper and let $g:Y\to S$ be separated. Then every $S$-morphism $h:X\to Y$ is proper. No Noetherian, reducedness or nonemptiness hypothesis is used, and the empty source is included. (Morphisms from a proper scheme to a separated one are proper)