= 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 divisor>[basepoint-free]. Its <Kodaira map> is
$$
[x_1,x_2,x_3,x_4]\longmapsto[x_1:x_3]\in\mathbb P^1.
$$
This 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.
Back to article page
