Intersection pairing on the Picard group of a surface ($\mathrm{Pic}(X)\times \mathrm{Pic}(X)\to \mathbb{Z}$)

On a smooth projective surface, two line bundles are represented by divisors $D$ and $E$, and their intersection is the degree of the zero-cycle obtained after moving them into proper position. It is a symmetric bilinear form on the Picard group, and $D\cdot E$ equals the degree of ${\mathcal{O}}_X(D)$ restricted to $E$ when $E$ is an integral curve not contained in $D$.

For the Hirzebruch surface ${\mathbb{F}}_n$, the negative section of a Hirzebruch surface $S$ and a fiber class of a Hirzebruch surface $F$ freely generate $\mathrm{Pic}({\mathbb{F}}_n)$ and satisfy