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DefineIf $D$ is a divisor class of degree zero, Riemann--Roch gives $\ell (D+Q)=1$, because $K_E\sim 0$ and the divisor $-D-Q$ has negative degree. A nonzero section therefore yieldsfor some point $P$, proving that $\alpha$ is surjective. If $\alpha (P)=\alpha (P^{\prime })$, then $P-P^{\prime }$ is a principal divisor. For $P\ne P^{\prime }$ its defining function would have one simple pole and give a degree-one map $E\to {\mathbb{P}}^1$, impossible for a genus-one curve. Thus $P=P^{\prime }$. The map and its inverse are regular, so this proves the Abel-Jacobi map of a genus-one curve is an isomorphism.

Transport the addition of the divisor class group through $\alpha$: define $P⊞R$ byThen $Q$ is the identity, and this is the chord-and-tangent group law on the plane cubic.

For the displayed equation, inversion is $(x,y)\mapsto (x,-y)$. Thus $P⊞P=Q$ exactly when $P=-P$. Besides the point at infinity $Q$, this requires $y=0$. Therefore the complete set of two-torsion points is

A standard sufficient hypothesis is that $X$ is both a Noetherian scheme and an integral scheme, and is regular in codimension one; in particular, a Noetherian normal scheme qualifies. A Weil divisor is then a finite sumover integral codimension-one closed subschemes $Z$. The local ring at the generic point of each $Z$ is a discrete valuation ring, so every nonzero rational function $g\in k(X{)}^{\times }$ has a principal divisorThe divisor class group is

A Weil divisor on $X$ is a finite formal sumover integral codimension-one closed subschemes $Z$. Since $X$ is regular in codimension one, the local ring at the generic point of each $Z$ is a discrete valuation ring. A nonzero rational function $f\in k(X{)}^{\times }$ therefore defines the principal divisor $\mathrm{div}(f)={\sum }_Z{v}_Z(f)[Z]$. The divisor class group is

The standard affine charts of ${\mathbb{P}}^3$ are copies of ${\mathbb{A}}^3$, and the product charts of ${\mathbb{P}}^1\times {\mathbb{P}}^1$ are copies of ${\mathbb{A}}^2$. Their local rings are localizations of polynomial rings over $k$ and hence are regular local rings. Both schemes are therefore regular.

On a regular integral scheme every Weil divisor is Cartier, so the divisor class group is naturally the Picard group. Pullback of line bundles along the Segre embedding therefore definesThe two groups areThe Segre coordinates are bihomogeneous of bidegree $(1,1)$, so ${\iota }^{\ast }{\mathcal{O}}_{{\mathbb{P}}^3}(1)=\mathcal{O}(1,1)$. In these bases the map is $m\mapsto (m,m)$ and

For a Noetherian separated integral scheme regular in codimension one, a Weil divisor is a finite integer combination of integral codimension-one closed subschemes. Principal divisors are the valuation divisors of nonzero rational functions, and the divisor class group is

For an open immersion $U\subseteq X$, restriction induces a surjectionevery prime divisor of $U$ closes to a prime divisor of $X$, while divisors supported in $X\setminus U$ form the kernel. Therefore $\mathrm{Cl}(X)=0$ implies $\mathrm{Cl}(U)=0$.

Affine schemes need not have trivial class group. For example,is a normal affine surface with $\mathrm{Cl}(X)\cong \mathbb{Z}/2\mathbb{Z}$. The height-one prime $(x,z)$ represents the nonzero class: twice this divisor is the principal divisor of $x$, but the divisor itself is not principal.

Let $D_1$ and $D_2$ be the toric divisors corresponding respectively to the primitive ray generators $v_1=(3,-2)$ and $v_2=(0,1)$ of the original cone. The principal divisor on a toric variety formula applied to the characters $x={\chi }^{e_1^{\ast }}$ and $y={\chi }^{e_2^{\ast }}$ givesThe toric divisor class sequence therefore presents the divisor class group asThe class of $D_1$ is a generator and $[D_2]=2[D_1]$.

For a smooth integral variety, every Weil divisor is a Cartier divisor, so the Picard group equals the divisor class group. If $U$ is obtained by removing irreducible Weil divisors $D_j$, the localization sequence for the divisor class group gives the displayed exact sequence. A rational trivialization of a line bundle on $U$ has divisor supported on the removed divisors, describing the kernel concretely.

On a normal integral Noetherian scheme, a nonzero rational function defines the displayed Weil divisor, with the sum over codimension-one prime divisors. The local ring at each such divisor is a discrete valuation ring, so the order is well defined. The divisor class group is the group of Weil divisors modulo principal Weil divisors. On a smooth curve this agrees with a principal divisor on an algebraic curve.