Self-intersection of the diagonal of a curve (${\Delta }_C^2=2-2g(C)$)

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 ${\Delta }_C^2=\mathrm{deg}TC=2-2g(C)$.