Normalized point blowups dominate local normal surface modifications

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

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-local-normal-surface-modification-dimension-and-projective-cohomology. Assume AC and DC. Let $(A,\mathfrak{m})$ be a normal Noetherian local domain of dimension two and $f:X\to \mathrm{Spec}A$ an integral modification. Then $X$ has dimension two, all closed points have local dimension two, $f$ is an isomorphism off the closed point, $f_{\ast }{\mathcal{O}}_X={\mathcal{O}}_{\mathrm{Spec}A}$, and its special fibre has dimension at most one. (Dimension and cohomology of local normal surface modifications)

lem-surface-modification-isomorphism-in-codimension-one. Assume AC. Let $f:X\to S$ be a modification of integral Noetherian schemes and let $S$ be normal of dimension two. Then $f$ is an isomorphism over an open subset containing every point of codimension at most one in $S$. The complement is a finite set of closed points. If every fibre is zero-dimensional, $f$ is an isomorphism. (A normal-surface modification is an isomorphism in codimension one)

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)

thm-one-dimensional-regular-local-rings-are-dvrs. Assume the Axiom of Choice (The Axiom of Choice). A nonzero Noetherian local ring of dimension one is regular if and only if it is a discrete valuation ring. Fields are excluded from the term DVR. (one dimensional regular local rings are dvrs)

thm-blowup-projective. Assume the Axiom of Choice, inherited from the relative Proj construction (The Axiom of Choice). Let $X$ be a scheme, let $\mathcal{I}$ be a quasi-coherent ideal sheaf of finite type on $X$ (def-quasi-coherent-ideal-sheaf) and let $\pi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{I}}X\to X$ be the blowup of def-blowup-scheme-along-ideal. Then: 1. (Blowups of finite type ideals are locally H-projective, and proper)

thm-pullback-center-ideal-invertible. Assume the Axiom of Choice, inherited from the relative Proj construction (The Axiom of Choice). Let $\mathcal{I}$ be a quasi-coherent ideal sheaf of finite type on a scheme $X$ (def-quasi-coherent-ideal-sheaf), let $\pi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{I}}X\to X$ be its blowup and let $E={\pi }^{-1}(Z)$ be the exceptional subscheme, with the convention that $\mathcal{O}(1)$ (The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier)

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-dimension-formula-for-affine-domains. Assume the Axiom of Choice. Let $k$ be a field, let $A$ be a finite-type $k$-domain, and let $\mathfrak{p}\in \mathrm{Spec}(A)$. Then $\mathrm{ht}(\mathfrak{p})+{\mathrm{trdeg}}_k\mathrm{Frac}(A\mathrm{/}\mathfrak{p})={\mathrm{trdeg}}_k\mathrm{Frac}(A)\mathrm{.}$ (The dimension formula for affine domains)

lem-blowup-of-closed-point-of-regular-surface-is-regular. Assume the Axiom of Choice (The Axiom of Choice). Let $S$ be a regular surface: a locally Noetherian scheme of pure dimension two all of whose local rings are regular (def-embedding-dimension-and-regular-local-ring). (Point blowups of regular surfaces stay regular, with rational exceptional curves at two-dimensional local rings)

lem-eventual-global-generation-coherent-twists. Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a Noetherian commutative ring (def-noetherian-ring-and-module) and let $X$ be a scheme projective over $A$ in the finite-dimensional H-projective convention (def-projective-morphism-pre-proj): the structure morphism $X\to \mathrm{Spec}A$ (def-affine-scheme-spectrum) factors over (Eventual generation of coherent projective twists)

A normal Noetherian domain satisfies $(R_1)$, so its height-one local rings are regular. (normal domain implies r one)