  In this article, we give a description of the closed cone of curves of the
projective bundle $\mathbb{P}(E)$ over a smooth projective variety $X$. Using
duality, we then calculate the nef cone of divisors in $\mathbb{P}(E)$ over
some special surfaces $X$ and for some special bundles on $X$. As an
application, we also calculate the Seshadri constants of semistable ample
vector bundles with vanishing discriminant on some special ruled surfaces at
special points.
