= Intersection pairing on the Picard group of a surface
{title2=$\operatorname{Pic}(X)\times\operatorname{Pic}(X)\to\mathbb Z$}
On a smooth projective surface, two <line bundle>[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 $D$.
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