Intersection number of a Cartier divisor with a curve ($D\cdot C$)

The intersection number $D\cdot C$ is the degree of the restriction of ${\mathcal{O}}_X(D)$ to the complete integral curve $C$. On a smooth surface, proper effective intersections are sums of local intersection multiplicities.

For a proper integral Cartier curve $C$ on a smooth algebraic surface, its self-intersection is $\mathrm{deg}{\mathcal{O}}_C(C)$. If $C$ is smooth, this is the degree of its normal line bundle. It is defined over any algebraically closed field, without requiring an associated real or complex manifold.

// Target: geometry-and-topology.bigb