On pseudoeffective cones of projective bundles and volume function

In this article, we compute the pseudo-effective cones of various projective bundles ${\mathbb{P}}_X(E)$ over higher dimensional varieties $X$ under some assumptions on $X$ as well as on the vector bundle $E$. We also compute the volume function on fibre product $\mathbb{P}(E){\times }_C\mathbb{P}(F)$ of two projective bundles over a smooth irreducible complex projective curve $C$. In particular, we show that the volume function on the fiber product of two ruled surfaces is of polynomial type.