Ruling morphism of the punctured three-dimensional affine quadric cone
= Ruling morphism of the punctured three-dimensional affine quadric cone
{title2=$\{xy=zw\}\setminus\{0\}\to\mathbb P^1$}
On $Y=\{xy=zw\}\setminus\{0\}$, the formulas
$$
[x:w]=[z:y]
$$
glue to a morphism $Y\to\mathbb P^1$. The inverse image of $[0:1]$ is the ruling plane $V(x,z)\cap Y$, so its divisor line bundle is the pullback of $\mathcal O_{\mathbb P^1}(1)$.