  Using the language of T-varieties, we study torus invariant curves on a
complete normal variety $X$ with an effective codimension-one torus action. In
the same way that the $T$-invariant Weil divisors on $X$ are sums of "vertical"
divisors and "horizontal" divisors, so too is each $T$-invariant curve a sum of
"vertical" curves and "horizontal" curves. We give combinatorial formulas that
calculate the intersection between $T$-invariant divisors and $T$-invariant
curves, and generalize the celebrated toric cone theorem to the case of
complete complexity-one $T$-varieties.
