On Zariski Decomposition with and without support

Abstract

In this paper we study Zariski Decomposition with support in a negative definite cycle, a variation introduced by Y. Miyaoka. We provide two extensions of the original statement, which was originally meant for effective $\Q$-divisors: we can either state it for any $\Q$-divisor, or we can take the support to be in any cycle. Ultimately, we present a new approach to Zariski Decomposition of pseudo-effective $\Q$-divisors, which consists in iterating Zariski Decomposition with support.

arXiv comment14 pages, a second improvement to Miyaoka's original result is added, along with a counterexample which shows that the two extensions are mutually exclusive; minor changes in the text structure, but the other results in the original posting of this paper remain unchanged