Real numerical divisor classes ($N^1(X{)}_{\mathbb{R}}$)

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 $N_1(X{)}_{\mathbb{R}}$.

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.

The ample cone consists of real numerical divisor classes represented by ample real divisors. It is an open convex cone, and Kleiman's criterion identifies it with the interior of the nef cone.

The sum of a nef real Cartier class and an ample real Cartier class on a projective scheme is ample. By Kleiman's criterion, the ample cone is the interior of the nef cone. If a ball about an ample class lies in that convex cone, translating it by a nef class still lies in the cone. The sum is consequently still an interior point.

The closed cone of curves is the closure in $N_1(X{)}_{\mathbb{R}}$ of the convex cone generated by classes of integral curves. A nef divisor pairs nonnegatively with this entire closed cone. Its boundary can contain limiting classes which are not represented by one curve.