Divisor class group ($\mathrm{Cl}(X)$)

The divisor class group is the group of Weil divisors modulo principal divisors of rational functions.

For positive $n,m$, $\mathrm{Cl}({\mathbb{P}}^n\times {\mathbb{P}}^m)={\mathbb{Z}}^2$, generated freely by the pullbacks of coordinate hyperplanes. Restrict a divisor to the standard affine chart, where unique factorization makes it principal, and subtract this principal Weil divisor to leave only the two boundary hyperplanes. Any principal relation between them would be given by a rational function whose divisor vanishes on the chart; unique factorization then makes that function a constant, proving independence. The same argument gives $\mathrm{Cl}({\mathbb{P}}^r)=\mathbb{Z}$.

If $C$ is an irreducible plane curve of degree $d$, the localization sequence for the divisor class group quotients $\mathrm{Cl}({\mathbb{P}}^2)=\mathbb{Z}[H]$ by the class $[C]=d[H]$. Nonsingularity of the removed curve is not required; irreducibility makes it a single removed prime Weil divisor.

For a smooth irreducible algebraic curve, send a divisor on an algebraic curve $D={\sum }_P{n}_{P}P$ to the line bundle associated to a divisor defined locally by ${\mathcal{O}}_X(D{)}_P=t_P^{-n_P}{\mathcal{O}}_{X,P}$. A nonzero rational section of a line bundle gives its inverse construction, and changing that section changes its divisor by a principal divisor. Thus the divisor class group is the Picard group. The quotient of the sheaf of nonzero rational functions on an irreducible variety by the sheaf of units of the structure sheaf has stalk $\mathbb{Z}$ at each closed point, via the discrete valuation. Its global sections are finite sums of points. The corresponding long exact sequence in sheaf cohomology, and the vanishing for the rational-function sheaf, identify both groups with $H^1(X,{\mathcal{O}}_X^{\ast })$.

For an elliptic curve over an algebraically closed field, ${\mathrm{Pic}}^0(E)\cong E(k)$ via $P\mapsto [P-O]$. This group is infinite and divisible by any prime $\ell$ different from the characteristic, since $[\ell ]:E\to E$ has nonzero derivative and is a surjective morphism of projective curves. A finitely generated abelian group divisible by $\ell$ has zero free rank and is finite. Hence $E(k)$, and therefore $\mathrm{Cl}(E)$, cannot be finitely generated. This proof applies to countable algebraically closed fields as well.

A Noetherian ring that is an integrally closed domain is a unique factorization domain exactly when its divisor class group vanishes. Indeed, vanishing says every prime Weil divisor, equivalently every height-one prime ideal, is principal.

Let $X$ be a Noetherian integral scheme that is regular in codimension one, and let $U=X\setminus {\bigcup }_{j=1}^r{Z}_j$, where the $Z_j$ are the codimension-one irreducible components of the complement. Restriction gives an exact sequence

If $A$ is a Noetherian normal domain and $S$ is a multiplicative set, then $\mathrm{Cl}(S^{-1}A)$ is the quotient of $\mathrm{Cl}(A)$ by the classes of height-one primes meeting $S$. Tracking the units of $S^{-1}A$ determines the relations among those prime divisors.

For the three-dimensional affine quadric conethe divisor class group is infinite cyclic. A generator is either ruling plane $V(x,z)$ or $V(x,w)$, and the principal divisor of $x$ gives the relation $[V(x,z)]+[V(x,w)]=0$.

For a Noetherian integral scheme $X$ that is regular in codimension one,where $H=X\times {\mathbb{P}}^{n-1}$ is a hyperplane divisor. Restriction to $X\times {\mathbb{A}}^n$ gives the first summand, while restriction to the generic projective-space fiber detects the coefficient of $H$.

For $n\ge 2$, the normal affine surfacehas $\mathrm{Cl}(X_n)\cong \mathbb{Z}/n\mathbb{Z}$. The class of the prime divisor $D=V(x,z)$ generates: localization at $x$ is a unique factorization domain, so the divisor-class localization sequence leaves only $D$, while $\mathrm{div}(x)=nD$ gives its exact order.