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.