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 $r\le e-1$ 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.