| A classical result in birational geometry, Mori's Cone Theorem, implies that if the canonical bundle of a variety X is not nef then X contains rational curves. This is the starting point of the so-called Minimal Model Program. In particular, hyperbolic varieties are positive, from the point of view of birational geometry. Very much in the same vein, one could ask what happens for a quasi projective variety, Y. Using resolution of singularity, then one is lead to consider pairs (X, D) of a variety and a divisor, such that Y=X\setminus D. I will show how to obtain a theorem analogous to Mori's Cone Theorem in this context. I will also discuss an ampleness criterion for hyperbolic pairs. |