The Picard group and intersection form of a product of projective lines

Given: a field $k$, the surface $X={\mathbb{P}}_k^1{\times }_k{\mathbb{P}}_k^1$ with its projections, the ruling fibres $\mathrm{\ell }={\mathrm{p}\mathrm{r}}_1^{-1}(\mathrm{\infty })$ and $m={\mathrm{p}\mathrm{r}}_2^{-1}(\mathrm{\infty })$, and the diagonal $\mathrm{\Delta }\subseteq X$.

By the structure lemma, $X$ is an integral smooth projective surface over $k$, so it is Noetherian, regular and (being smooth of finite type over $k$) locally factorial; the projections ${\mathrm{p}\mathrm{r}}_1,{\mathrm{p}\mathrm{r}}_2$ are flat and proper, and $\pi :{\mathbb{P}}_k^1\to \mathrm{Spec}k$ is flat (The product of two projective lines is an integral smooth projective surface, Regular local rings are unique factorization domains, Locally factorial scheme, Flat morphism of schemes). The intersection product on $X$ is defined, symmetric and $\mathbb{Z}$-bilinear (Intersection numbers of Cartier divisors on a smooth projective surface, The surface intersection product is symmetric and bilinear).

Pullbacks of Cartier divisors along flat morphisms are defined: a regular section pulls back to a regular section under a flat morphism, so the pullback datum of Pullback of a Cartier divisor exists; and for a Cartier divisor $D$ one has ${\mathcal{O}}_X(f^{\ast }D)\cong f^{\ast }{\mathcal{O}}_Y(D)$ (Pullback of a Cartier divisor, Pullback of a Cartier divisor computes the pullback of its line bundle, Flat morphism of schemes). The point $\mathrm{\infty }\in {\mathbb{P}}_k^1$ is an effective Cartier divisor with ${\mathcal{O}}_{{\mathbb{P}}^1}(\mathrm{\infty })\cong {\mathcal{O}}_{{\mathbb{P}}^1}(1)$: the coordinate $x_0$ is a nonzero global section of $\mathcal{O}(1)$ (Global sections of projective twists) with zero scheme $\mathrm{\infty }$ (A regular global section of an invertible sheaf glues to an effective Cartier divisor, Relative projective space from standard charts).

Intersection with a curve is the degree of the restriction: for an effective Cartier divisor $C$ and any Cartier divisor $D$, $C\cdot D={\mathrm{deg}}_C({\mathcal{O}}_X(D){\mathrm{\mid }}_C)$, where ${\mathrm{deg}}_C(\mathcal{N})=\chi (C,\mathcal{N})-\chi (C,{\mathcal{O}}_C)$ (Intersection with a curve is the degree of the restriction, Degree of an invertible sheaf on a proper one-dimensional scheme, Euler characteristic of a coherent sheaf). On ${\mathbb{P}}_k^1$ one has $h^0(\mathcal{O})=1$, $h^0(\mathcal{O}(1))=2$ and $h^1(\mathcal{O})=h^1(\mathcal{O}(1))=0$ (Global sections of projective twists, Top cohomology of projective twists), so ${\mathrm{deg}}_{{\mathbb{P}}^1}({\mathcal{O}}_{{\mathbb{P}}^1}(1))=1$.

Class groups: $X$ is a locally factorial Noetherian integral scheme, so $\mathrm{Pic}(X)\cong \mathrm{Cl}(X)$ compatibly with the Weil divisor of a Cartier divisor, and $\mathrm{CaDiv}(X)\mathrm{/}{\mathrm{Prin}}_C(X)\cong \mathrm{Pic}(X)$ (Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme, On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group, Weil divisor normal noetherian scheme, Principal weil divisor and class group).

Excision: for $U=X\setminus (\mathrm{\ell }\cup m)$ the restriction of Weil divisors (closure in $X$ of each prime divisor of $U$) is surjective with kernel $\mathbb{Z}[\mathrm{\ell }]\oplus \mathbb{Z}[m]$, and principal divisors restrict to principal divisors; hence $\mathrm{Cl}(X)\to \mathrm{Cl}(U)$ is surjective with kernel generated by $[\mathrm{\ell }],[m]$ (Weil divisor normal noetherian scheme, Principal weil divisor and class group, Order codimension one rational function, Height-one localizations of normal Noetherian domains are DVRs). Moreover $U={\mathrm{p}\mathrm{r}}_1^{-1}(U_0)\cap {\mathrm{p}\mathrm{r}}_2^{-1}(V_0)=U_0{\times }_k{V}_0$ for the standard affine charts of the two factors, and $U_0{\times }_k{V}_0\cong \mathrm{Spec}k[y_1]{\times }_k\mathrm{Spec}k[y_2]\cong \mathrm{Spec}k[y_1,y_2]$ (Relative projective space from standard charts, Affine fibre products are spectra of tensor products, Projective space is Proj of a polynomial ring). To verify the class-group kernel, a prime divisor meeting $U$ has the same codimension-one local ring at its generic point on $U$ and on the whole scheme, so restriction preserves its valuation and the divisor of every rational function. Prime divisors of $U$ extend by closure, proving surjectivity. If a divisor restricts to ${\mathrm{div}}_U(f)$, use the same $f$ in the common function field and subtract its divisor on the whole scheme; the difference is supported on the removed prime divisors. Conversely those boundary divisors restrict to zero. This proves the asserted exactness.

In $k[y_1,y_2]$ every height-one prime is principal and every irreducible element is prime, and every Weil divisor is a finite combination of prime divisors, so every Weil divisor on $U\cong \mathrm{Spec}k[y_1,y_2]$ is principal: for a combination ${\sum }_i{n}_i[V(g_i)]$ the rational function ${\prod }_i{g}_i^{n_i}$ has divisor ${\sum }_i{n}_i[V(g_i)]$, since each $g_i$ has order one along $V(g_i)$ and order zero along the other primes (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes, Unique factorisation domain, Order codimension one rational function, Discrete valuations). Hence $\mathrm{Cl}(U)=0$.

Diagonal: the pullbacks ${\mathrm{p}\mathrm{r}}_1^{\ast }{x}_j$ and ${\mathrm{p}\mathrm{r}}_2^{\ast }{y}_j$ of the coordinate sections are global sections of ${\mathrm{p}\mathrm{r}}_1^{\ast }\mathcal{O}(1)$ and ${\mathrm{p}\mathrm{r}}_2^{\ast }\mathcal{O}(1)$ whose zero schemes are the fibres ${\mathrm{p}\mathrm{r}}_1^{-1}(V(x_j))$ and ${\mathrm{p}\mathrm{r}}_2^{-1}(V(y_j))$; the section $s={\mathrm{p}\mathrm{r}}_1^{\ast }{x}_0\otimes {\mathrm{p}\mathrm{r}}_2^{\ast }{y}_1-{\mathrm{p}\mathrm{r}}_1^{\ast }{x}_1\otimes {\mathrm{p}\mathrm{r}}_2^{\ast }{y}_0$ of the invertible sheaf ${\mathrm{p}\mathrm{r}}_1^{\ast }\mathcal{O}(1)\otimes {\mathrm{p}\mathrm{r}}_2^{\ast }\mathcal{O}(1)$ is nonzero, and on the two equal-index charts its coefficient is the difference of the affine coordinates, while on a mixed chart it is, up to sign, $1-tu$. The equal-index equations identify the coordinates; on a mixed chart $tu=1$ identifies the two projective points on their overlap. Thus its zero scheme is the diagonal $\mathrm{\Delta }$ (Zero scheme of a line-bundle section, A regular global section of an invertible sheaf glues to an effective Cartier divisor, Pullback of a Cartier divisor computes the pullback of its line bundle, Relative projective space from standard charts). Consequently $\mathrm{\Delta }$ is an effective Cartier divisor with ${\mathcal{O}}_X(\mathrm{\Delta })\cong {\mathrm{p}\mathrm{r}}_1^{\ast }\mathcal{O}(1)\otimes {\mathrm{p}\mathrm{r}}_2^{\ast }\mathcal{O}(1)$.

The Axiom of Choice is inherited from the intersection, pullback and class-group suppliers above; all computations use the two ruling fibres, the diagonal and finitely many chart functions.