Self-intersection number ($[X]\cdot [X]$)

For an oriented submanifold of half the dimension of its ambient manifold, the self-intersection number is the intersection number with a transverse perturbation. It equals the Euler number of the normal bundle.

For a smooth projective algebraic curve $C$, the normal bundle of its diagonal in $C\times C$ is $TC$: on the diagonal the quotient of $TC\oplus TC$ by the diagonal subbundle is identified with $TC$ by taking the difference. The self-intersection formula therefore gives ${\Delta }_C^2=\mathrm{deg}TC=2-2g(C)$.