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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.

The nef cone is the closed convex cone of nef divisor classes. It is dual to the closed cone of curves. Its interior is the ample cone on a projective scheme.

Take $V=C$ for each integral projective curve in $X$. The assumed positivity gives $D\cdot C>0$, hence certainly $D\cdot C\ge 0$. By the definition of a nef divisor,This is only the curve test. It does not already prove ampleness: positivity against each individual curve can fail to be uniform on limiting classes in the closed cone of curves. The hypotheses in higher dimensions, used in the next sections, are what rule out this failure.

The Nakai–Moishezon criterion says that a Cartier divisor $D$ on a projective scheme is ample exactly whenfor every positive-dimensional integral closed subvariety $Y$. Kleiman's criterion says that the ample cone is the interior of the nef cone; equivalently, the numerical class of $D$ is ample exactly when it is strictly positive on every nonzero element of the closed cone of curves $\overline{\mathrm{NE}}(X)$. Positivity merely on individual curves is insufficient: the closure of the cone is essential. A nef divisor has nonnegative intersection number with every integral curve, and its restriction to every closed subscheme is nef.