= Self-intersection number
{title2=$[X]\cdot[X]$}
= Self-intersection
{synonym}
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 that
$$
[S]\cdot[S]=\langle e(N_{S/M}),[S]\rangle.
$$
The <Euler class> makes the answer independent of the chosen transverse <section of a fiber bundle>.<Characteristic Classes by John Milnor and James D. Stasheff>[Chapter 11, Theorem 11.3: restriction of the dual class to the submanifold is the normal Euler class]{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.
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