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I was reading the wikipedia page on intersection number: http://en.wikipedia.org/wiki/Intersection_number#Intersection_multiplicities_for_plane_curves

and I was confused by property number 6 they want there. I cannot justify why this is a natural property to have.

I have seen the definition of intersection number via dimension of the complete local ring of the scheme theoretic intersection over the base field. From this definition property 6 is natural, but I would like to see a low-brow (i.e. in classical language), intuitive reason for expecting 6.

Thanks!

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    I think my explanation requires that both $P$ and $Q$ are irreducible at $(x(t_0),y(t_0))$ in sense of germs of holomorphic functions, and you can use property 5 to reduce any case to that special case. In your example, $x^2=x\cdot x$, so all you need is to parametrise $x=0$, calculate the intersection number of $y=0$ and $x=0$ then double it.2012-05-15

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The point is that the system of equations $P = Q = 0$ is equivalent to $P + RQ = Q = 0$, meaning you can solve one if and only if you can solve the other. Another point of view is that the intersection multiplicity of $P$ and $Q$ at $p$ should only depend on the ideal $I = (P,Q)$, suitably localized to only look at what is happening near $p$ (and not at other intersection points). But on the level of ideals, $(P,Q) = (P + RQ, Q)$. This last equality is really what my first sentence is expressing more loosely.

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    Ah, I see your point now. Thanks! That clears my doubts.2012-05-30