Linear systems on blow-ups of Hirzebruch surfaces

Motivated by various equivalent versions of the SHGH conjecture for $\mathbb P^2$ blown up at very general points, we propose a similar conjecture for Hirzebruch surfaces.

We prove that this conjecture is true for the Hirzebruch surface $\mathbb F_e$ blown up at $r\leqslant e+5$ very general points.