Blowing up a regular point is a contraction

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

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-blowup-scheme-along-ideal. Assume the Axiom of Choice as inherited from the relative Proj construction (The Axiom of Choice). Let $X$ be a scheme and let $\mathcal{I}\subseteq {\mathcal{O}}_X$ be a quasi-coherent ideal sheaf of finite type (def-quasi-coherent-ideal-sheaf), with zero scheme $Z=V(\mathcal{I})$, the closed subscheme of $X$ cut out by $\mathcal{I}$. (Blowup of a scheme along an ideal sheaf)

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)

def-integral-scheme. An integral scheme is a nonempty scheme that is reduced and whose underlying topological space is irreducible. Equivalently, it is nonempty and every nonempty affine open is the spectrum of a domain. The latter criterion is independent of the chosen affine open cover. (Integral schemes)

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)

cor-blowup-birational-integral-scheme. Assume the Axiom of Choice, inherited from the blowup construction (The Axiom of Choice). Let $X$ be an integral scheme (Integral schemes) and let $\mathcal{I}$ be a nonzero quasi-coherent ideal sheaf of finite type. (Blowing up a nonzero ideal on an integral scheme is birational)

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-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-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)