Let E be a vector bundle of rank r over a smooth projective variety X, and let \(\pi :Y=\mathbb {P}(E)\rightarrow X\) be the associated projective bundle of one-dimensional quotients. We show that if the relative anticanonical divisor \(-K_{Y/X}\) is nef (or equivalently E is universally semistable in the sense of Fulger and Langer (J Algebra 609:657–687, 2022), then the rth self intersection \((K_{Y/X})^r\) is numerically trivial. Using this, we show that if \(-K_{Y/X}\) is nef, then nef (resp. pseudoeffective) cycles of codimension c on Y are precisely those of the form \(\begin{aligned} \sum _{i=0}^{r-1} (-K_{Y/X})^i \cdot \pi ^* \alpha _i \end{aligned}\) where \(\alpha _i\) are nef (resp. pseudoeffective) cycles of codimension \(c-i\) on X. We also describe the nef/pseudoeffective cycles on Y when E is certain extension of a universally semistable bundle. Our results generalize the key propositions of Fulger (Math Z 269:449–459, 2011) about pseudoeffective cycles on projective bundles over curves. Even when specialized to nef or pseudoeffective divisors, our results are cleaner and more general than some recent computation of Misra et al. (Osaka J Math 59:639–651, 2022; On pseudoeffective cones of projective bundles and volume function, preprint https://arxiv.org/abs/2203.07007).
