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

Let $S$ be a compact, oriented, boundaryless embedded submanifold of dimension $n$ in an oriented smooth manifold $M$ of dimension $2n$. Its self-intersection number is the signed intersection number of $S$ with a small transverse push-off. A transverse section of a fiber bundle of the normal bundle gives that push-off; its signed zeros show thatThe Euler class makes the answer independent of the chosen transverse section of a fiber bundle.^[ref]

A line in the Complex projective plane has square $1$. A fiber ${\mathbb{P}}^1\times \{p\}$ in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ has square zero because it has a disjoint homologous push-off. The exceptional divisor of a blowup of a smooth algebraic surface has square $-1$. For noncompact embedded submanifolds or embedded submanifolds with boundary, an integer requires additional support or boundary data; the unrestricted signed-count definition does not apply.

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