The work described in my talk is related to one of the key problems in higher dimensional geometry, namely the existence of what is called a Zariski decomposition for divisors on a smooth projective variety. It was shown in pioneering work by O. Zariski and D. Mumford that such a decomposition exists on surfaces, and that it has fundamental consequences for understanding linear systems and for classification problems. Unfortunately on higher dimensional varieties such a decomposition does not exist. The purpose of the work I am describing is partly to develop methods which would overcome its absence.