Seshadri constants on blow-ups of Hirzebruch surfaces

Let $e,r \ge 0$ be integers and let $\mathbb{F}_e : = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(-e))$ denote the Hirzebruch surface with invariant $e$. We compute the Seshadri constants of an ample line bundle at an arbitrary point of the $r$-point blow-up of $\mathbb{F}_e$ when $r \leq e-1$ and at a very general point when $r=e$ or $r=e+1$. We also discuss several conjectures on linear systems of curves on the blow-up of $\mathbb{F}_e$ at $r$ very general points.