Local normalized point sequences spread at closed surface points

Given: A normal integral surface $X$ locally of finite type over a permitted base, a closed point $x\in X$ with two-dimensional local ring $B$, and a finite sequence of normalized point blowups over $\mathrm{Spec}B$.

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-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)

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)

lem-relative-spec-glues-affine-algebras. Let $S$ be a scheme and let $\mathcal{A}$ be an affine-locally module-associated sheaf of commutative unital ${\mathcal{O}}_S$-algebras, as in def-affine-local-quasi-coherent-algebra. Put $B_U=\mathrm{\Gamma }(U,\mathcal{A})$ for each affine open $U\subseteq S$. (Glue relative spectra of affine-local algebras)

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-blowup-base-change-flat. Assume the Axiom of Choice as inherited from the relative Proj construction. Let $g\text{\hspace{0.17em}⁣}:X^{\mathrm{\prime }}\to X$ be a flat morphism of schemes and $\mathcal{I}$ a quasi-coherent ideal sheaf of finite type on $X$. (Flat base change for blowups, and failure without flatness)

thm-integrality-commutes-with-localisation. Let $A\to B$ be a homomorphism of commutative rings, let $S\subseteq A$ be multiplicative, and let $b\in B$. 1. If $b$ is integral over $A$, then $b\mathrm{/}1$ is integral over $S^{-1}A$ in $S^{-1}B$. 2. If $b\mathrm{/}1$ is integral over $S^{-1}A$ in $S^{-1}B$, then some $s\in S$ makes $sb$ integral over $A$. (Integrality and integral closure commute with localisation)