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I have a theorem in the book Complex Algebraic Curves- Frances Kirwan :

*If two projective curves $C$ and $D$ of degrees $n$ and $m$ respectively in $P_2$ intersect at exactly $n^2$ points and if $nm$ points if these lie on an irreducible curve $E$ of degree $m < n$ then the remaining $n(n-m)$ points of intersection lie on a curve of degree at most $n-m$.

My question is do we count the multiplicity of intersection while counting the $n^2$ points?

For instance if we have a point of intersection of multiplicity $2$, do we count the point once or twice?

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  • $\begingroup$ Yes: these points are counted with multiplicities. $\endgroup$ – Crostul Mar 30 '16 at 10:38
  • $\begingroup$ @Crostul thanks!! $\endgroup$ – inquisitive Mar 30 '16 at 10:43

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