Hirzebruch surface (${\mathbb{F}}_k$)

The Hirzebruch surface is the ruled surfaceIt is the smooth toric surface with rays $(1,0),(0,1),(-1,k),(0,-1)$.

The negative section $S\subset {\mathbb{F}}_k$ has self-intersection $S^2=-k$. Together with a fiber $F$ of the ruling, it generates the Picard group and satisfies $F^2=0$ and $S\cdot F=1$.

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.