Real 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.

A real Cartier divisor is ample if it is a positive real combination of ample Cartier divisors. Equivalently its numerical class lies in the ample cone. To recover an actual positive combination from the numerical condition, write the divisor in a finite Cartier basis and take nearby rational points in the inverse image of the open ample cone. A small rational simplex around the original coefficient vector expresses it as a positive convex combination of rational ample Cartier combinations; clearing denominators gives ample Cartier divisors.

Real linear equivalence means that $D-E$ is a finite real combination of principal Cartier divisors. Rational linear equivalence uses rational coefficients instead. Both imply numerical equivalence of divisors.