Curves on Hirzebruch Surfaces and Semistability

When $a\ge2$, 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.