  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.
