The Picard group of a point blowup of the projective plane

Given: a field $k$, the projective plane $X={\mathbb{P}}_k^2$ with its twisting sheaves ${\mathcal{O}}_X(d)$, a $k$-rational point $p\in X(k)$, the blowup $\pi :X^{\mathrm{\prime }}={\mathrm{Bl}}_{p}X\to X$ with exceptional curve $E={\pi }^{-1}(p)$, and the class $\mathrm{\ell }={\pi }^{\ast }{\mathcal{O}}_X(1)$.

The plane: $X$ is an integral regular projective surface over $k$ of pure dimension two. Its standard charts are the affine planes $\mathrm{Spec}k[y_1,y_2]$ with coordinate rings polynomial domains of dimension two and regular local rings, the charts are Noetherian and form a finite cover, and $X$ is projective over $k$ hence proper (Projective space is Proj of a polynomial ring, Relative projective space from standard charts, A polynomial ring in finitely many indeterminates over an integral domain is an integral domain, A polynomial ring in n variables over a field has dimension n, Dimension can be computed on an open cover, localisation and polynomial extension of regular rings, Integral schemes, Intersection numbers of Cartier divisors on a smooth projective surface, Projective morphisms before Proj, Projective morphisms are proper). As a smooth finite-type $k$-scheme, $X$ is locally factorial (Regular local rings are unique factorization domains, Locally factorial scheme).

Blowup calculus at a $k$-rational point: the residue degree is $r=[\kappa (p):k]=1$; $X^{\mathrm{\prime }}$ is an integral regular projective surface over $k$; $E$ is an effective Cartier divisor with $E\cong {\mathbb{P}}_k^1$ and ${\mathcal{O}}_E(E)\cong {\mathcal{O}}_{{\mathbb{P}}_k^1}(-1)$, so $E\cdot E=-r=-1$; and for all Cartier divisors $D,D^{\mathrm{\prime }}$ on $X$ one has $E\cdot {\pi }^{\ast }D=0$ and ${\pi }^{\ast }D\cdot {\pi }^{\ast }{D}^{\mathrm{\prime }}=D\cdot D^{\mathrm{\prime }}$ (The intersection matrix of a point blowup of a regular surface, Exceptional subscheme of a blowup, Intersection numbers of Cartier divisors on a smooth projective surface).

Cohomology of twists on the plane: $H^0(X,{\mathcal{O}}_X(m))$ is the degree-$m$ part of $k[x_0,x_1,x_2]$ for $m\ge 0$ and vanishes for $m<0$; $H^1(X,{\mathcal{O}}_X(m))=0$ for every $m$; $H^2(X,{\mathcal{O}}_X(m))=0$ for $m>-3$ (Global sections of projective twists, Intermediate cohomology of projective twists vanishes, Top cohomology of projective twists). Hence $\chi (X,{\mathcal{O}}_X)=1$, $\chi (X,{\mathcal{O}}_X(-1))=0$ and $\chi (X,{\mathcal{O}}_X(-2))=0$ (Euler characteristic of a coherent sheaf).

Lines are effective Cartier divisors with associated sheaf ${\mathcal{O}}_X(1)$: a nonzero linear form is a global section of ${\mathcal{O}}_X(1)$ (Global sections of projective twists), it is regular on the integral scheme $X$, and its zero scheme is an effective Cartier divisor $L$ with ${\mathcal{O}}_X(L)\cong {\mathcal{O}}_X(1)$ (A regular global section of an invertible sheaf glues to an effective Cartier divisor, Invertible sheaf of cartier divisor).

Total transform: for a reduced effective Cartier divisor $C$ on the regular surface $X$ with a closed point $p$ at which the multiplicity $m={\mathrm{mult}}_p(C)$ is finite and positive, ${\pi }^{\ast }C=C^{\mathrm{\prime }}+mE$ as effective Cartier divisors, where $C^{\mathrm{\prime }}$ is the strict transform (Total transform equals strict transform plus multiplicity times the exceptional divisor, Strict transform of a closed subscheme); passage to associated invertible sheaves uses ${\mathcal{O}}_{X^{\mathrm{\prime }}}({\pi }^{\ast }C)\cong {\pi }^{\ast }{\mathcal{O}}_X(C)$ (Pullback of a Cartier divisor computes the pullback of its line bundle, Pullback of a Cartier divisor).

Divisors and classes: a Weil divisor on a Noetherian normal scheme is a finite integral combination of prime divisors, the principal Weil divisor of $f\in K(X{)}^{\times }$ is ${\mathrm{div}}_W(f)={\sum }_Z{\mathrm{ord}}_Z(f)[Z]$, and $\mathrm{Cl}$ is the quotient by principal divisors (Weil divisor normal noetherian scheme, Principal weil divisor and class group, Order codimension one rational function). The order along a prime divisor with generic point $\xi$ is the valuation of the discrete valuation ring ${\mathcal{O}}_{X,\xi }$, which is a DVR because $X$ is normal and Noetherian (Height-one localizations of normal Noetherian domains are DVRs, Discrete valuations); orders are additive, vanish on units, and a uniformiser has order one.

On a locally factorial Noetherian integral scheme, Cartier and Weil divisors agree compatibly with principal divisors, and the canonical map $\mathrm{Pic}(X)\to \mathrm{Cl}(X)$ carrying $[{\mathcal{O}}_X(D)]$ to the class of the associated Weil divisor is an isomorphism (Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme); the map $D\mapsto [{\mathcal{O}}_X(D)]$ induces an isomorphism $\mathrm{CaDiv}(X)\mathrm{/}{\mathrm{Prin}}_C(X)\stackrel{\sim }{\to }\mathrm{Pic}(X)$ (On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group, Invertible sheaf of cartier divisor).

Off the centre: $\pi$ restricts to an isomorphism ${\pi }^{-1}(X\setminus \{p\})\to X\setminus \{p\}$ (The blowup is an isomorphism off the center), and ${\pi }^{-1}(X\setminus \{p\})=X^{\mathrm{\prime }}\setminus E$ (Exceptional subscheme of a blowup). For a reduced effective Cartier divisor $L$ through $p$ with strict transform $m$, this identifies $m\setminus E$ with $L\setminus \{p\}$ (Strict transform of a closed subscheme).

Standard charts: after a linear change of homogeneous coordinates taking the line $L$ to $V(x_0)$, $X\setminus L$ is the affine chart $U_0=\mathrm{Spec}k[x_1\mathrm{/}{x}_0,x_2\mathrm{/}{x}_0]\cong {\mathbb{A}}^2$, a polynomial domain over $k$ (Relative projective space from standard charts, Projective space is Proj of a polynomial ring, A polynomial ring in finitely many indeterminates over an integral domain is an integral domain).

In a polynomial ring $k[y_1,y_2]$ every height-one prime is principal, generated by an irreducible element, and every irreducible element is prime (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes, Unique factorisation domain). Consequently every Weil divisor on $\mathrm{Spec}k[y_1,y_2]$ is principal: a prime divisor is $V(g)$ for an irreducible $g$, and for a finite combination ${\sum }_i{n}_i[V(g_i)]$ the rational function ${\prod }_i{g}_i^{n_i}$ has divisor ${\sum }_i{n}_i[V(g_i)]$, because each $g_i$ has order one along $V(g_i)$ and order zero along the other $V(g_j)$ by [F6], [F10].

Excision for class groups: if $Z$ is a prime divisor on an integral Noetherian normal scheme $X$ with $U=X\setminus Z$, restriction of Weil divisors (closure in $X$ of each prime divisor of $U$) is surjective with kernel $\mathbb{Z}[Z]$, and principal divisors restrict to principal divisors; hence the induced map $\mathrm{Cl}(X)\to \mathrm{Cl}(U)$ is surjective and its kernel is the image of $\mathbb{Z}[Z]$ (Weil divisor normal noetherian scheme, Principal weil divisor and class group, Order codimension one rational function). The same statement holds with $Z$ replaced by a union of finitely many prime divisors, with kernel the direct sum of their classes. 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.

Intersection numbers depend only on the linear equivalence classes (Intersection numbers of Cartier divisors on a smooth projective surface, The surface intersection product is symmetric and bilinear, Cartier divisor); in particular a Cartier divisor linearly equivalent to $0$ has zero intersection with every Cartier divisor.

A nonzero global section of an invertible sheaf on the integral scheme $X$ is regular with effective Cartier zero scheme (A regular global section of an invertible sheaf glues to an effective Cartier divisor); duals and tensor products of invertible sheaves are invertible and ${\mathcal{L}}^{\vee }\otimes \mathcal{L}\cong {\mathcal{O}}_X$ (Dual of a line bundle is its tensor inverse, Invertible sheaves).

The Axiom of Choice is inherited from the projective-space, blowup and divisor-class suppliers above; all selections below are finite (a line, a point, finitely many prime divisors).