The intersection matrix of a point blowup of a regular surface

Given: a field $k$, an integral regular projective surface $X$ over $k$, a closed point $p\in X$ with residue field $\kappa (p)$ and residue degree $r=[\kappa (p):k]$, the blowup $\pi :X^{\mathrm{\prime }}={\mathrm{Bl}}_{p}X\to X$ and the exceptional curve $E={\pi }^{-1}(p)$, and the Axiom of Choice (The Axiom of Choice).

Blowup interfaces: $X^{\mathrm{\prime }}={\mathrm{Bl}}_{I}X={\mathrm{Proj}}_{X}R(I)$ for the ideal sheaf $I$ of the closed point $p$, with structural morphism $\pi$ and relative twists (Blowup of a scheme along an ideal sheaf); $E={\pi }^{-1}(p)$ is the scheme-theoretic inverse image of the center, a closed subscheme with ideal $I{\mathcal{O}}_{X^{\mathrm{\prime }}}$ (Exceptional subscheme of a blowup); $\pi$ is an isomorphism over $X\setminus p$ and $E$ is the complement of that open subscheme (The blowup is an isomorphism off the center); and $\pi$ is proper and locally H-projective, and globally H-projective as soon as the ideal is generated by finitely many global sections (Blowups of finite type ideals are locally H-projective, and proper).

Integrality and regularity of $X^{\mathrm{\prime }}$: since $X$ is integral and $I$ is a nonzero ideal of finite type, $X^{\mathrm{\prime }}$ is integral and $\pi$ is birational (Blowing up a nonzero ideal on an integral scheme is birational); and since $X$ is a regular finite-type $k$-scheme of pure dimension two and $p$ is a closed point, $X^{\mathrm{\prime }}$ is regular of pure dimension two, $E$ is an effective Cartier divisor isomorphic to ${\mathbb{P}}_{\kappa (p)}^1$, and ${\mathcal{O}}_E(E)\cong {\mathcal{O}}_{{\mathbb{P}}_{\kappa (p)}^1}(-1)$ (Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field); alternatively $E\cong \mathbb{P}(I\mathrm{/}{I}^2)={\mathbb{P}}_{\kappa (p)}^1$ for the regular immersion $p\hookrightarrow X$ (Regular centers have projective-bundle exceptional divisors, The normal bundle of the exceptional curve is O(-1)).

Projective embeddings: an ample twist of the coherent point ideal on $X$ is globally generated (Eventual generation of coherent projective twists, Global generation by the evaluation map). The graded algebra $⨁I^d\otimes L^{\otimes d}$ has the same relative Proj as $R(I)$ for invertible $L$ (Invariance of the blowup under invertible (fractional) rescaling of the ideal). A graded quotient of ${\mathcal{O}}_X[z_0,\dots ,z_N]$ gives a closed subscheme of ${\mathbb{P}}_X^N$ (Relative Proj of a graded quasi-coherent algebra, Closed subschemes of projective space and saturated ideals). Closed immersions are preserved by base change (Closed immersions are affine quotients and survive base change). The closed point of a finite-type scheme over $k$ has finite residue degree (A maximal ideal of an affine algebra has finite residue field over the base field).

Cohomology of line bundles on the exceptional curve: for $E\cong {\mathbb{P}}_{\kappa (p)}^1$ over $\kappa (p)$ and an invertible sheaf $\mathcal{M}$ on $E$ of degree $d$ over $\kappa (p)$, ${\chi }_k(E,\mathcal{M})=r(1+d)$, where $r=[\kappa (p):k]$; in particular ${\chi }_k(E,{\mathcal{O}}_E)=r$, and for ${\mathcal{O}}_E(E)\cong \mathcal{O}(-1)$ of degree $-1$ one has ${\chi }_k(E,{\mathcal{O}}_E(E))=0$ and ${\mathrm{deg}}_E({\mathcal{O}}_E(E))={\chi }_k(E,{\mathcal{O}}_E(E))-{\chi }_k(E,{\mathcal{O}}_E)=-r$ (Euler characteristic of line bundles on a projective line over a finite field extension, Degree of an invertible sheaf on a proper one-dimensional scheme).

Pushforward and projection formula: ${\pi }_{\ast }{\mathcal{O}}_{X^{\mathrm{\prime }}}={\mathcal{O}}_X$ and $R^q{\pi }_{\ast }{\mathcal{O}}_{X^{\mathrm{\prime }}}=0$ for every $q>0$ (Pushforward and vanishing for point blowups on a surface); consequently $\chi (X^{\mathrm{\prime }},{\pi }^{\ast }\mathcal{N})=\chi (X,\mathcal{N})$ for every invertible ${\mathcal{O}}_X$-module $\mathcal{N}$ (Projection formula for invertible twists, Euler characteristic of a coherent sheaf).

Effective divisors, twists and transforms: for an effective Cartier divisor $H$ on a surface, the sequence $0\to \mathcal{N}(-H)\to \mathcal{N}\to i_{\ast }(\mathcal{N}{\mathrm{\mid }}_H)\to 0$ is exact for invertible $\mathcal{N}$ (Effective Cartier divisors give a short exact sequence, Twisting the exact sequence of an effective Cartier divisor); the total transform ${\pi }^{\ast }D$ of a Cartier divisor is the pullback Cartier divisor with ${\mathcal{O}}_{X^{\mathrm{\prime }}}({\pi }^{\ast }D)\cong {\pi }^{\ast }{\mathcal{O}}_X(D)$ (Total transform of a Cartier divisor, Pullback of a Cartier divisor, Pullback of a Cartier divisor computes the pullback of its line bundle); and for a reduced effective Cartier divisor $C$ on the regular surface $X$ through $p$ with multiplicity $m\ge 1$, the total transform decomposes as ${\pi }^{\ast }C=C^{\mathrm{\prime }}+mE$ with $C^{\mathrm{\prime }}$ the strict transform, which is reduced (Total transform equals strict transform plus multiplicity times the exceptional divisor, Strict transform of a closed subscheme, The reduction of a scheme).

The intersection product of Intersection numbers of Cartier divisors on a smooth projective surface is symmetric and $\mathbb{Z}$-bilinear on the Picard group of an integral regular projective surface, and the restriction theorem identifies $H\cdot F={\mathrm{deg}}_H(\mathcal{O}(F){\mathrm{\mid }}_H)$ for effective $H$ (The surface intersection product is symmetric and bilinear, Intersection with a curve is the degree of the restriction); degrees, and with them intersection numbers, are additive on the proper curve $H$ (Degree is additive on invertible sheaves over a proper curve).

The Axiom of Choice enters through the blowup, sheaf-cohomology, global-generation and Euler-characteristic suppliers above; the points and divisors appearing below are given data.