Self-intersection of an algebraic curve ($C^2=\mathrm{deg}{\mathcal{O}}_C(C)$)

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