The number of contracted curves drops by one after factoring through a point blowup

Given: A field $k$, integral regular finite-type proper $k$-schemes $X,Y$ of pure dimension two, a birational morphism $f\text{\hspace{0.17em}⁣}:X\to Y$, a closed point $y\in Y$ such that $f$ factors as $f=b\circ g$ through the blowup $b\text{\hspace{0.17em}⁣}:Y^{\mathrm{\prime }}={\mathrm{Bl}}_{y}Y\to Y$, and the finite number $N(f)$ of integral curves of $X$ contracted by $f$.

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-finite-morphism-schemes. A morphism of schemes $f:X\to S$ is finite if, for every affine open $U=\mathrm{Spec}A\subseteq S$, its inverse image is affine, $f^{-1}(U)=\mathrm{Spec}B$, and the induced $A$-algebra $B$ is module-finite over $A$ in the sense of def-finite-type-and-module-finite-algebras. The affineness language agrees with def-affine-morphism-schemes. (Finite morphisms of schemes)

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 (Quasi-finite morphisms of 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-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-finite-birational-to-normal-is-isomorphism. Assume the Axiom of Choice. Let $f\text{\hspace{0.17em}⁣}:Y\to X$ be a finite birational morphism of irreducible classical varieties over an algebraically closed field. If $X$ is normal, then $f$ is an isomorphism. The normality of the target is essential: the normalization of the cusp is finite and birational but not an isomorphism. (A finite birational morphism onto a normal variety is an isomorphism)

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

A proper birational morphism of integral Noetherian schemes with normal two-dimensional target is an isomorphism over every codimension-one point. (A normal-surface modification is an isomorphism in codimension one)