Blowing up a smooth point: charts, exceptional curve, and contraction

Given: A field $k$, an integral regular finite-type $k$-scheme $S$ of pure dimension two, a $k$-rational closed point $p\in S$, and the blowup $\pi \text{\hspace{0.17em}⁣}:S^{\mathrm{\prime }}={\mathrm{Bl}}_{p}S\to S$ with exceptional curve $E={\pi }^{-1}(p)$.

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-exceptional-curve-and-contraction. Assume the Axiom of Choice where it is inherited from the degree and intersection suppliers below (The Axiom of Choice). Let $X$ be a Noetherian scheme. (a) Exceptional curves of the first kind. A closed subscheme $E\subseteq X$ (def-closed-immersion-schemes) is an exceptional curve of the first kind if: 1. (Exceptional curves of the first kind and their contractions)

lem-blowing-up-a-regular-point-is-a-contraction. Assume the Axiom of Choice. Assume the Axiom of Choice, inherited from the cited blowup and intersection suppliers. 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-blowup-intersection-matrix-at-smooth-point. Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field, let $X$ be an integral regular projective surface over $k$ (def-divisor-intersection-number-on-smooth-projective-surface), let $p\in X$ be a closed point with residue field $\kappa (p)$ and $r:=[\kappa (p):k]$, let $\pi :X^{\mathrm{\prime }}={\mathrm{Bl}}_{p}X\to X$ be the blowup of $X$ at $p$. (The intersection matrix of a point blowup of a regular surface)

lem-blowup-isomorphism-off-center. Let $\mathcal{I}$ be a quasi-coherent ideal sheaf of finite type with zero scheme $Z$ and let $\pi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{I}}X\to X$ be the blowup. (The blowup is an isomorphism off the center)

lem-exceptional-curve-normal-bundle-minus-one. Assume the Axiom of Choice. Let $p$ be a closed point of a regular surface $S$ over a field $k$, assume $\dim {\mathcal{O}}_{S,p}=2$, and let $\pi \text{\hspace{0.17em}⁣}:S^{\mathrm{\prime }}\to S$ be the blowup of $p$ and $E$ its exceptional curve. (The normal bundle of the exceptional curve is O(-1))

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)

The affine blowup along $I=(x,y)$ is covered by $A[I\mathrm{/}x]$ and $A[I\mathrm{/}y]$, whose overlap inverts $T=y\mathrm{/}x$ and $U=x\mathrm{/}y$ with $TU=1$. (Affine blowup standard charts and overlaps)

thm-intersection-with-curve-as-degree-of-restriction. Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field, let $X$ be an integral regular projective surface over $k$ (def-divisor-intersection-number-on-smooth-projective-surface) and let $C$ and $D$ be effective Cartier divisors on $X$ (def-effective-cartier-divisor, def-cartier-divisor) with associated line bundles ${\mathcal{O}}_X(C)$ and (Intersection with a curve is the degree of the restriction)