Ample Cartier divisor

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

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.

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 a projective scheme, a divisor class is in the ample cone exactly when it is strictly positive on every nonzero element of the closed cone of curves. Equivalently, the ample cone is the interior of the nef cone. The projectivity assumption matters: the same characterization is not asserted here for arbitrary proper schemes.

An integral Cartier divisor $D$ on a projective scheme is ample if and only if $D^{\dim Y}\cdot Y>0$ for every positive-dimensional integral closed subvariety $Y$. Testing only curves is sufficient in dimension one, but not in higher dimensions.

A real Cartier class on a projective scheme is ample if and only if its top self-intersection on every positive-dimensional integral subvariety is positive. For the converse curve tests give nefness. Small ample perturbations and rational approximation give ample rational $B,C$ with $D-A=B-C$ satisfying the algebraic Morse inequality for ample divisors, hence bigness. Induction gives ample restrictions on codimension-one subvarieties; uniform ample subtraction from a big divisor with ample exceptional restrictions makes $D-ϵA$ nef. The nef-plus-ample ampleness lemma concludes. For reducible schemes perform the finite component tests simultaneously.