The normal bundle of the exceptional curve is O(-1)

Given: A regular surface $S$ over $k$, a closed point $p\in S$ with two-dimensional local ring, the blowup $\pi \text{\hspace{0.17em}⁣}:S^{\mathrm{\prime }}\to S$ of $p$ with exceptional curve $E$ and $A={\mathcal{O}}_{S,p}$.

Choice. The Axiom of Choice is assumed, as in the statement; the cited suppliers used below are stated under it (The Axiom of Choice).

Regular centers have projective-bundle exceptional divisors: For a closed point $p$ of a regular surface $S$ over a field $k$ with two-dimensional local ring, the conormal sheaf $\mathcal{I}\mathrm{/}{\mathcal{I}}^2={\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2$ is free of rank two over $\kappa (p)$ and the exceptional divisor is isomorphic to the projective line ${\mathbb{P}}_{\kappa (p)}^1$ over $\kappa (p)$; the corollary identifies it with the projective bundle ${\mathbb{P}}_Z(\mathcal{I}\mathrm{/}{\mathcal{I}}^2)$ in the quotient convention.

Affine blowup standard charts and overlaps, Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, Flat base change for blowups, and failure without flatness and regular local rings are domains and cohen macaulay: The localized point blowup has charts $A[(x,y)\mathrm{/}x]$ and $A[(x,y)\mathrm{/}y]$ with inverse ratio overlap, where $x,y$ are regular parameters and form a regular sequence in the domain $A$.

Projective bundle in the quotient convention: The projective bundle of a finite locally free module $E$ of rank $r$ over $S$ is the relative Proj ${\mathbb{P}}_S(E)={\mathrm{Proj}}_S\mathrm{Sym}(E)\to S$, in the quotient convention in which an $S$-morphism $T\to {\mathbb{P}}_S(E)$ amounts to an isomorphism class of surjections $g^{\ast }E\to L$ with $L$ invertible on $T$.

The twist index on the projective line is an isomorphism invariant: For the twists of the relative projective line over a field, ${\mathcal{O}}_{{\mathbb{P}}_k^1}(n)\cong {\mathcal{O}}_{{\mathbb{P}}_k^1}(m)$ if and only if $n=m$; hence the twist index attached to an invertible sheaf on ${\mathbb{P}}_k^1$ is an isomorphism invariant.

Twisting sheaf on Proj: For a commutative nonnegatively graded ring $S$, the twisting sheaf on $\mathrm{Proj}S$ is ${\mathcal{O}}_X(n)=\widetilde{S(n)}$, the associated sheaf of the shifted graded module, with $\mathrm{\Gamma }(D_{+}(f),{\mathcal{O}}_X(n))=S(n{)}_{(f)}$ on standard opens.

The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier: For the blowup of a quasi-coherent ideal sheaf $\mathcal{I}$ of finite type with exceptional subscheme $E=V(\mathcal{I}{\mathcal{O}}_{\mathrm{Bl}})$: $\mathcal{O}(1)$ is invertible, the inverse-image ideal $\mathcal{I}{\mathcal{O}}_{\mathrm{Bl}}$ is invertible and equals $\mathcal{O}(1)$, and the exceptional divisor is effective Cartier with ${\mathcal{O}}_{\mathrm{Bl}}(-E)=\mathcal{I}{\mathcal{O}}_{\mathrm{Bl}}=\mathcal{O}(1)$ and ${\mathcal{O}}_{\mathrm{Bl}}(E)=\mathcal{O}(-1)$.