The intersection pairing on the projective plane

Given: a field $k$, the projective plane $X={\mathbb{P}}_k^2$ with twisting sheaves $\mathcal{O}(d)$, a line $l$, and nonzero homogeneous forms $f,g$ of degrees $d,e\ge 1$ with zero schemes $C=Z(f)$, $D=Z(g)$.

${\mathbb{P}}_k^2$ is an integral regular projective surface over $k$ of pure dimension two: its affine charts are spectra of polynomial domains in two variables over $k$, whose local rings are regular and whose dimension is two (localisation and polynomial extension of regular rings, Affine-domain dimension equals transcendence degree), and the irreducible charts form a pairwise intersecting open cover (Relative projective space from standard charts, Projective space is Proj of a polynomial ring, Integral schemes, Intersection numbers of Cartier divisors on a smooth projective surface). Its twisting sheaves are invertible and satisfy $\mathcal{O}(m)\otimes \mathcal{O}(n)\cong \mathcal{O}(m+n)$ and $\mathcal{O}(-d)\cong \mathcal{O}(d{)}^{\vee }$ (Invertible twists for degree-one generated rings, Dual of a line bundle is its tensor inverse).

Cohomology of twists: on ${\mathbb{P}}_k^2$ the cohomology of $\mathcal{O}(m)$ vanishes except in degrees $0$ and $2$, with $H^0\cong k[x_0,x_1,x_2{]}_m$ for $m\ge 0$ (dimension $\left(\genfrac{}{}{0px}{}{m+2}{2}\right)$), $H^0=0$ for $m<0$, and $H^2$ described by the negative Laurent monomials, nonzero exactly for $m\le -3$ with dimension $\left(\genfrac{}{}{0px}{}{-m-1}{2}\right)$ (Cohomology of O(d) on projective space, Relative projective space from standard charts). Consequently the Euler characteristic of Euler characteristic of a coherent sheaf satisfies $\chi (X,\mathcal{O}(m))=\left(\genfrac{}{}{0px}{}{m+2}{2}\right)=\frac{(m+2)(m+1)}{2}\text{for every }m\in \mathbb{Z}\mathrm{.}$

Regular sections and divisors: on the integral scheme $X$ a nonzero global section of an invertible sheaf is regular, so its zero scheme is an effective Cartier divisor $Z(s)$ with ${\mathcal{O}}_X(Z(s))\cong \mathcal{L}$ (A regular global section of an invertible sheaf glues to an effective Cartier divisor, Zero scheme of a line-bundle section, Effective cartier divisor, Invertible sheaf of cartier divisor). For the nonzero form $f$ of degree $d\ge 1$ the section $f\in \mathrm{\Gamma }(X,\mathcal{O}(d))$ is nonzero and its zero scheme is $C$; likewise for $g$.

The restriction-degree theorem (Intersection with a curve is the degree of the restriction): for effective Cartier divisors $C,D$ on the surface $X$ one has $C\cdot D={\mathrm{deg}}_C(\mathcal{O}(D){\mathrm{\mid }}_C)$, where ${\mathrm{deg}}_C$ is the Euler-characteristic degree of Degree of an invertible sheaf on a proper one-dimensional scheme. For the line $l=Z(x_0)\subseteq X$, which by [F3] is an effective Cartier divisor with associated line bundle ${\mathcal{O}}_X(l)\cong \mathcal{O}(1)$, with closed immersion $j:l\hookrightarrow X$, twisting the exact sequence of Twisting the exact sequence of an effective Cartier divisor gives $0\to \mathcal{O}(m-1)\to \mathcal{O}(m)\to j_{\ast }(\mathcal{O}(m){\mathrm{\mid }}_l)\to 0$ for every $m\in \mathbb{Z}$; additivity of $\chi$ on short exact sequences of coherent modules (Euler characteristic is additive in short exact sequences) and the closed-immersion projection formula $\chi (X,j_{\ast }\mathcal{F})=\chi (l,\mathcal{F})$ for coherent $\mathcal{F}$ on $l$ (Projection formula for a closed immersion and an invertible sheaf) therefore give $\chi (l,\mathcal{O}(m){\mathrm{\mid }}_l)=\chi (X,\mathcal{O}(m))-\chi (X,\mathcal{O}(m-1))=m+1$ by [F2], in particular ${\mathrm{deg}}_l(\mathcal{O}(2){\mathrm{\mid }}_l)=3-1=2$.

The intersection product on the integral regular projective surface $X$ is the symmetric $\mathbb{Z}$-bilinear pairing of The surface intersection product is symmetric and bilinear, defined by the alternating sum of Intersection numbers of Cartier divisors on a smooth projective surface.

The Axiom of Choice enters through the cohomology and Euler-characteristic suppliers of [F2] and the curve-degree supplier of [F4]; no selection is made below.