Contractible classes in toric varieties

Let X be a smooth, complete, toric variety. We study those curves C in X that are contractible, in the sense that there exists an equivariant morphism with connected fibers, with source X, that contracts exactly the irreducible curves that are numerically equivalent to a multiple of C. When X is projective, we compare contractible and extremal curves, and we show that every curve in X is numerically equivalent to a linear combination with positive integral coefficients of contractible curves.