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The Abel map of an algebraic curve sends an effective Cartier divisor $D$ to its line bundle ${\mathcal{O}}_C(D)$. Its scheme fibre over a line bundle $L$ is the projective space of one-dimensional subspaces of $H^0(C,L)$. This follows in families: a relative effective Cartier divisor in the fixed class determines a nowhere fibre-zero section of $L$ up to a line bundle from the parameter space, equivalently a line subbundle of $H^0(C,L)\otimes {\mathcal{O}}_T$. Thus the scheme fibre is smooth, not merely reduced on its points. The dimensions can nevertheless jump, so the Abel map of an algebraic curve need not be a smooth morphism.

For ample rational Cartier classes $B,C$ on an integral projective $n$-fold with $n\ge 1$, $B^n>n{B}^{n-1}\cdot C$ implies $B-C$ is big. Scale to very ample integral divisors. Choose an effective Cartier divisor $G\in |C|$. Repeated restriction gives $h^0(m(B-C))\ge h^0(mB)-m{h}^0(G,mB)$: multiply each negatively twisted restriction by a section avoiding its associated points to inject it into ${\mathcal{O}}_G(mB)$. Asymptotic Riemann–Roch and Serre vanishing give the positive leading lower bound along sufficiently divisible section indices after undoing the scaling. No complex-analytic Morse theory or characteristic-zero vanishing is used.

A Cartier divisor is ample when its associated line bundle is ample. The Nakai–Moishezon criterion and Kleiman's criterion characterize this condition numerically.

For a Cartier divisor $D$ on an $n$-dimensional projective scheme, $\chi (X,{\mathcal{O}}_X(mD))$ is a polynomial in $m$ whose leading term is $D^n{m}^n/n!$. The intersection product uses the fundamental cycle, with the generic multiplicity of each component. Thus the statement also applies to nonreduced schemes.

A Cartier divisor $D$ on an integral $n$-dimensional projective variety is big when $h^0(X,mD)\ge c{m}^n$ for some $c>0$ and infinitely many positive integers $m$. Equivalently, its Iitaka dimension is $n$. Kodaira's lemma characterizes bigness by an ample-plus-effective decomposition, and the birational linear system criterion for bigness shows why this growth captures the full variety.

Let $\nu :X^{\nu }\to X$ be the finite normalization of an integral projective variety. The coherent sheaf $\mathcal{Q}={\nu }_{\ast }{\mathcal{O}}_{X^{\nu }}/{\mathcal{O}}_X$ is supported in dimension at most $n-1$. For a Cartier divisor $D$, the projection formula for sheaves and the resulting long exact sequence in sheaf cohomology giveThe last step uses the polynomial bound for sections of a fixed divisor. Therefore the leading order $m^n$ growth, and hence bigness, is preserved in both directions. This handles nonnormal varieties without assuming a resolution of singularities in positive characteristic.

A Cartier divisor on an integral projective variety is big if and only if some multiple has a complete linear system of a divisor giving a rational map birational onto its image. Kodaira's lemma embeds a very ample subsystem in a suitable multiple. Conversely, $n$ algebraically independent elements among the section ratios give $\left(\genfrac{}{}{0ex}{}{q+n}{n}\right)$ independent degree-$q$ section monomials, where $n=\dim X$. Smoothness is not necessary.

A canonical divisor $K_X$ on a smooth integral variety is the Cartier divisor of a nonzero rational top-degree differential form; ${\mathcal{O}}_X(K_X)\cong {\omega }_X$. Its linear-equivalence class is independent of the chosen form. The plurigenus is the dimension of the space of sections of a positive tensor power of this bundle.

On an irreducible variety, a Cartier divisor is a section of ${\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$: its local rational equations differ by regular units. Their ratios define an invertible sheaf, and a global rational equation gives the trivial class. Conversely a rational trivialization of an invertible sheaf gives local equations. The rational-function sheaf is flasque, so its long exact sequence in sheaf cohomology also identifies the quotient with $H^1(X,{\mathcal{O}}_X^{\ast })$.

For an integral domain $A$ with fraction field $K$, realize Cartier divisors as invertible fractional ideals, with local divisor equation $f$ corresponding to the ideal $fA$. This is the $\mathcal{O}(-D)$ convention. The maps in the displayed exact sequence are unit inclusion, $f\mapsto fA$, and $I\mapsto [I]$. A principal fractional ideal is trivial as a divisor exactly when its generator is a unit. An invertible fractional ideal is trivial in the Picard group of a ring exactly when it is principal. Every rank-one projective module embeds in $K$ and produces an invertible fractional ideal, proving surjectivity. Local multiplication identifies $I{\otimes }_{A}J$ with $IJ$, so these are group homomorphisms.

The toric divisor class sequence identifies the divisor class group with $\mathbb{Z}$ via $(a_i)\mapsto \sum q_i{a}_i$. Choose integers $a_i$ with $\sum q_i{a}_i=1$; then $D=\sum a_i{F}_i$ represents its positive generator $H$. This is an ample Q-Cartier divisor: for $\ell =\mathrm{lcm}(q_i)$ the local Cartier data of $\ell D$ on the chart omitting ray $j$ is $m_j=(\ell /q_j)e_j-\ell (a_i)$, which satisfies the strict toric ampleness criterion. The generator need not itself be a Cartier divisor; for example the positive generator on $\mathbb{P}(1,1,2)$ has Cartier index two.

This is the full linear system of divisors, obtained by using every global section of the associated line bundle. For a Cartier divisor on an integral projective variety, the complete linear system $|D|$ consists of its effective representatives under linear equivalence of Cartier divisors. A basis of $H^0(X,{\mathcal{O}}_X(D))$ defines a rational map to projective space.

Choose an effective Cartier divisor $E$ of degree at least $g$ on a smooth projective genus-$g$ algebraic curve. For a family $M$ of degree-$g-1$ line bundles, the evaluation map from ${\pi }_{\ast }M(E)$ to ${\pi }_{\ast }(M(E){|}_E)$ is between vector bundles of the same rank. Its determinant vanishes exactly when $H^0(C,M)$ is nonzero. Changing $E$ changes the presentation by cancelling an invertible determinant factor; the zero Cartier divisor is the theta divisor. At a point with $h^0(M)=h^1(M)=1$, the derivative is the cup-product map $H^1({\mathcal{O}}_C)\to \mathrm{Hom}(H^0(M),H^1(M))$. By Serre duality it is nonzero, since the product of nonzero sections of $M$ and $K_C\otimes M^{-1}$ is nonzero. Hence this Cartier divisor is generically reduced.

On an integral scheme, for a Cartier divisor locally represented by $a_i\in K^{\ast }$, its divisor line bundle is the subsheaf of rational functions locally equal to $a_i^{-1}{\mathcal{O}}_X$. Unit ratios glue these free rank-one modules. Given a nonzero rational section of a line bundle $s=a_i{e}_i$, the maps $e_i\mapsto a_i^{-1}$ identify that invertible sheaf with ${\mathcal{O}}_X(D)$. Replacing $s$ by a nonzero rational multiple changes $D$ by a principal Cartier divisor.

For an effective Cartier divisor $E$ and any Cartier divisor $D$, multiplication by a defining section of $E$ gives the displayed short exact sequence of sheaves. Its long exact sequence in sheaf cohomology compares sections and cohomology on $X$ with their restrictions to $E$.

A Cartier divisor $D$ on a projective scheme is ample exactly when $\chi (V,{\mathcal{O}}_V(mD))\to +\infty$ for every positive-dimensional integral closed subvariety $V$, including irreducible components. For the converse induct on dimension: restrictions to hyperplane sections are ample, so higher cohomology vanishing from an ample hyperplane restriction gives $h^0=\chi +h^1\to \infty$. A nonzero section vanishing at a chosen point then exists. The vanishing-section ampleness criterion finishes. Divergence alone does not imply a positive top-degree coefficient.

For an integral domain $A$ with fraction field $K$, a fractional ideal is a nonzero $A$-submodule $I\subseteq K$ for which $dI\subseteq A$ for some $0\ne d\in A$. A finitely generated submodule of $K$ has a common denominator and is therefore a fractional ideal. The product is generated by pairwise products of elements. An invertible fractional ideal has a multiplicative inverse and is finitely generated; arbitrary fractional ideals need not be finitely generated over a non-Noetherian domain. The familiar fractional ideals of a number field are the special case $A={\mathcal{O}}_K$. This general definition also applies to the ideals representing Cartier divisors on affine integral schemes.