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For a coherent sheaf $\mathcal{F}$ of support dimension $d$ on a projective scheme and a fixed Cartier divisor $D$, $h^0(X,\mathcal{F}(mD))=O(m^d)$. Choose a sufficiently high ample divisor $A$ whose section avoids the associated points of $\mathcal{F}$ and for which $D+A$ is ample. Multiplication by the $m$th power of that section injects $\mathcal{F}(mD)$ into $\mathcal{F}(m(D+A))$. The ample Hilbert polynomial, together with Serre vanishing, bounds the latter section space by $O(m^d)$.

The positivity of a Cartier divisor can be measured by sections, intersection numbers, or its numerical class. Ample divisors give projective embeddings after taking a multiple; nef divisors are their numerical limits; big divisors have the maximum possible order of section growth. Nefness and bigness are different conditions.

For a proper morphism of varieties $f:X\to Y$, a Cartier divisor $D$ on $Y$, and an algebraic cycle $\alpha$ on $X$, pullback of $D$, intersection with $\alpha$, and proper pushforward satisfy the displayed identity in the Chow group. In particular, for a generically finite morphism of surfaces of degree $r$, $(f^{\ast }D{)}^2=r{D}^2$. For a curve $C$ contracted to a point, $f_{\ast }C=0$, so $f^{\ast }D\cdot C=0$.

For a morphism of varieties $f:X\to Y$, pull back local defining rational functions of a Cartier divisor $D$ whenever they remain nonzero on each component of $X$. Their ratios remain regular units, defining $f^{\ast }D$. Equivalently, pull back the associated line bundle and its rational section. A dominant morphism always permits this construction.

A Weil divisor is Q-Cartier if some positive integer multiple is a Cartier divisor. Its Cartier index is the smallest such integer. It is ample in the Q-Cartier sense when such a multiple is an ample Cartier divisor. This need not make the original divisor itself Cartier.

On an integral scheme with generic point $\eta$ and function field $K$, a rational section of a line bundle $\mathcal{L}$ is an element of the one-dimensional $K$-space ${\mathcal{L}}_{\eta }$. A nonzero rational section writes as $a_i{e}_i$ in local trivializations, with $a_i\in K^{\ast }$. Ratios of these coefficients are regular units, and therefore define a Cartier divisor.

A real Cartier divisor is a finite real linear combination of Cartier divisors. A rational Cartier divisor is defined analogously with rational coefficients; after multiplication by a common denominator it is Cartier.

The finite-dimensional real vector space $N^1(X{)}_{\mathbb{R}}$ consists of real combinations of Cartier divisors modulo numerical equivalence of divisors. Its elements are numerical divisor classes. The space of curve classes paired with it is $N_1(X{)}_{\mathbb{R}}$.

For a Cartier divisor on a smooth projective surface, the displayed formula computes its Euler characteristic of a coherent sheaf. Combined with Serre duality, it turns intersection numbers into lower bounds for section spaces.

A Cartier divisor is semiample if some positive multiple is a basepoint-free divisor, equivalently its divisor line bundle has a globally generated positive power. Such a power defines a Kodaira map $f$ with $\mathcal{O}(mD)=f^{\ast }\mathcal{O}(1)$. Semiampleness implies nefness but need not imply ampleness: a fibre divisor of a morphism to a curve is a basic example.

A smooth algebraic surface is a two-dimensional smooth variety. Its local rings are regular, and its integral curves define Cartier divisors.

A smooth projective surface is a two-dimensional smooth variety admitting a closed embedding in projective space. Its integral curves are Cartier divisors, and their intersections, arithmetic adjunction and point blowups control its birational geometry.

The scheme $C^{(d)}$ parametrizes effective degree-$d$ Cartier divisors on a smooth algebraic curve. Near a Cartier divisor $\sum m_i{P}_i$, its completed local ring is a power-series ring in the elementary symmetric coordinates of $m_i$ local parameters at each $P_i$. It is therefore smooth of dimension $d$, even when the characteristic of a field divides $d!$. For projective $C$ it is projective.

The theta divisor consists of degree-$g-1$ line bundles with a nonzero global section. It is the irreducible image of the symmetric product of a curve $C^{(g-1)}$, with its reduced Cartier divisor structure. The determinant description of the theta divisor supplies equations, multiplicities and pullbacks in families.

Evaluation determinants for $L\otimes P$ and $L$ have a quotient which is a rational section of ${⨂}_i{\mathrm{pr}}_i^{\ast }P$, not intrinsically a scalar. If $P^N\cong {\mathcal{O}}_C$, its $N$th power becomes a rational function after choosing that trivialization. In Cartier divisor notation, write $P={\mathcal{O}}_C(p)$ and $\mathrm{div}(h)=Np$. With raw meromorphic evaluations the scalar expression is $r^N{\prod }_{i}h(z_i)$. The extra product accounts for the line-bundle frames; it is suppressed when evaluation already uses compatible torsion trivializations.

A Cartier divisor is ample if for every positive-dimensional integral closed subvariety some positive multiple restricts to a bundle with a nonzero section having a nonempty zero locus. Inductively the divisor is ample on all lower-dimensional subschemes. On an integral component the chosen section cuts out a nonempty effective Cartier divisor $E$ whose restriction bundle ${\mathcal{O}}_E(E)$ is ample. The restriction ampleness implies semiampleness for an effective divisor lemma makes the original divisor semiample. On a curve the vanishing section forces positive degree, so the semiample and curve-positive ampleness criterion proves ampleness.