Ruling morphism of the punctured three-dimensional affine quadric cone ($\{xy=zw\}\setminus \{0\}\to {\mathbb{P}}^1$)

On $Y=\{xy=zw\}\setminus \{0\}$, the formulasglue 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 ${\mathcal{O}}_{{\mathbb{P}}^1}(1)$.