Fiber class of a Hirzebruch surface ($F$)

The fiber class $F$ is the divisor class of a fiber of ${\mathbb{F}}_k\to {\mathbb{P}}^1$. Its complete linear system gives the ruling.

For a toric fiber divisor $F$ on ${\mathbb{F}}_1$,The bundles $\mathcal{O}(kF)$ are basepoint-free for $k\ge 0$ but never ample, since $F^2=0$.