Factorization of birational morphisms of regular surfaces into point blowups

Given: A field $k$, integral regular finite-type proper $k$-schemes $X,Y$ of pure dimension two, and a birational morphism $f\text{\hspace{0.17em}⁣}:X\to Y$.

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-birational-morphism-schemes. Let $k$ be a field and let $X$ and $Y$ be integral $k$-schemes of finite type (def-integral-scheme, def-locally-finite-type-and-finite-type-morphism). (Birational morphisms of integral finite-type schemes)

def-etale-locus-morphism. Let $f:X\to S$ be a morphism locally of finite presentation (def-locally-finite-presentation-morphism). The étale locus of $f$ is $\mathrm{Et}(f)=\{x\in X:\text{the morphism }f\text{ is }\acute{\text{e}}\text{tale at }x\}\subseteq X,$ the set of points at which the conditions of def-etale-morphism-schemes hold. (def-etale-locus-morphism)

def-proper-morphism. A morphism of schemes $f:X\to S$ is proper if and only if it is separated, of finite type, and universally closed. Here separatedness has the meaning of def-separated-morphism-schemes, finite type has the meaning of def-locally-finite-type-and-finite-type-morphism, and universally closed has the meaning of def-universally-closed-morphism. (Proper morphisms)

def-quasi-finite-morphism-schemes. A morphism of schemes $f:X\to S$ is quasi-finite if it is of finite type (def-locally-finite-type-and-finite-type-morphism) and, for every point $x\in X$, there are affine neighbourhoods $U=\mathrm{Spec}(B)$ of $x$ and $V=\mathrm{Spec}(A)$ of $f(x)$ such that $f(U)\subseteq V$ and the induced finite-type ring map $A\to B$ is quasi-finite at the prime (def-quasi-finite-morphism-schemes)

def-smooth-morphism-schemes. Let $f:X\to S$ be a morphism of schemes, let $x\in X$ and put $s=f(x)$. The morphism $f$ is smooth at $x$ when the following three conditions hold at $x$: 1. $f$ is locally of finite presentation at $x$ (def-locally-finite-presentation-morphism); 2. $f$ is flat at $x$ (def-flat-morphism-schemes); 3. (def-smooth-morphism-schemes)

lem-birational-surface-morphism-factors-through-blowup-at-a-non-isomorphism-point. Assume the Axiom of Choice. Let $k$ be a field, let $X$ and $Y$ be integral regular finite-type $k$-schemes of pure dimension two that are proper over $k$, and let $f\text{\hspace{0.17em}⁣}:X\to Y$ be a birational morphism. (A birational morphism of regular surfaces factors through the blowup of a point where its inverse is undefined)

lem-blowing-up-a-regular-point-is-a-contraction. Assume the Axiom of Choice. Let $k$ be a field, let $S$ be an integral regular finite-type $k$-scheme of pure dimension two, let $p\in S$ be a closed point, let $\pi \text{\hspace{0.17em}⁣}:S^{\mathrm{\prime }}={\mathrm{Bl}}_{p}S\to S$ be the blowup of $S$ at $p$ and let $E={\pi }^{-1}(p)$ be its exceptional curve. (Blowing up a regular point is a contraction)

lem-contracted-curve-count-decreases-under-a-point-blowup-factorization. Assume the Axiom of Choice. Let $k$, $X$, $Y$ and $f\text{\hspace{0.17em}⁣}:X\to Y$ be as in A birational morphism of regular surfaces factors through the blowup of a point where its inverse is undefined, and let $N(f)$ be the number of integral curves of $X$ contracted by $f$, a nonnegative integer by Fibres of a proper birational morphism of regular surfaces. (The number of contracted curves drops by one after factoring through a point blowup)

lem-fibre-components-of-a-proper-birational-morphism-of-regular-surfaces. Assume the Axiom of Choice. Let $k$ be a field, let $X$ and $Y$ be integral regular finite-type $k$-schemes of pure dimension two, let $f\text{\hspace{0.17em}⁣}:X\to Y$ be a proper birational morphism and let $y\in Y$ be a closed point. Put $F=f^{-1}(y)$. Then: 1. $F$ is a proper $\kappa (y)$-scheme with $\dim F\le 1$. 2. (Fibres of a proper birational morphism of regular surfaces)

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)

thm-blowup-regular-surface-closed-point-regular. Assume the Axiom of Choice. Let $S$ be a regular finite-type $k$-scheme of pure dimension two, let $p$ be a closed point, put $\kappa =\kappa (p)$ and $r=[\kappa :k]$, and let $\pi \text{\hspace{0.17em}⁣}:S^{\mathrm{\prime }}={\mathrm{Bl}}_{p}S\to S$ be the blowup of $S$ at $p$ with exceptional subscheme $E$. (Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field)