On the existence of a weak Zariski decomposition on projectivized vector bundles

A pseudoeffective divisor is said to have a weak Zariski decomposition if it can be, up to a birational transformation, numerically written as the sum of a nef and an effective divisor. In this paper we consider the problem of the existence of a weak Zariski decomposition for each pseudoeffective divisor on a variety $$X= \mathbb {P}(\fancyscript{E})$$ , where $$\fancyscript{E}$$ is a vector bundle on a smooth complex projective variety $$Z$$ of Picard number one. We prove the existence of such a decomposition in a number of meaningful situations.