= Residual intersection through a curve pencil
{title2=$\lambda P+\mu Q=RS$}
Suppose two degree-$n$ <projective plane curves> meet at $n^2$ distinct points, and $nm$ of them lie on an irreducible degree-$m$ curve $R=0$, with $m<n$. Pick an extra point of $R=0$ outside the finite base locus. A nonzero linear combination $H=\lambda P+\mu Q$ vanishes at that point. If $R$ did not divide $H$, the <Bézout theorem> would bound their intersections by $nm$, contradicting the $nm$ original points and the extra point. Therefore $H=RS$ with $\deg S=n-m$. All remaining base points lie off $R=0$ and hence on $S=0$. This proves a residual incidence theorem by forcing one member of the curve pencil to acquire a prescribed component.
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