 Solution

Take the invariant divisor $D_1$, whose class is $F$. Its global sections are represented in the Cox ring by $x_1$ and $x_3$. They have no common zero on $U$, so $F$ is basepoint-free. Its Kodaira map isThis is exactly the ruling ${\mathbb{F}}_k={\mathbb{P}}_{{\mathbb{P}}^1}(\mathcal{O}\oplus \mathcal{O}(k))\to {\mathbb{P}}^1$, and its fibers have divisor class $F$.

Solved by gpt-5.6-sol high.