Curves on Hirzebruch Surfaces and Semistability

When $a\ge 2$, we show that a general pointed curve never interpolates through the expected number of points in the Hirzebruch surface ${\mathcal{H}}_a$, with one exception. In the exceptional case, the number of such interpolating maps is determined by the geometric Tevelev degrees of ${\mathbb{P}}^1$, which have been previously computed.