 Big cone

The big cone in $N^1(X{)}_{\mathbb{R}}$ is open and convex: write a big class as ample plus effective, perturb only the ample summand, and use openness of the ample cone. For positive-dimensional $X$, intersection with $H^{n-1}$ for any very ample divisor $H$ is strictly positive on every big class. Hence this cone contains no line and does not contain the zero class.