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There is the theorem that if $P$ is a simple point on a plane curve $F$, then for any plane curves $G,H$ we have $I(P,F\cap(G+H))\geq\min(I(P,F\cap G),I(P,F\cap H))$.

I need to find a counter-example to this inequality when the premise that $P$ is simple does not hold. I'm finding this very hard. At the moment I'm simply picking curves, $F$ which has a multiple point at the origin and some $G,H$ I think I can work with, and either the inequality still holds, or I can't complete the calculation because they're too difficult for the methods I know. I'm not really sure why the fact $P$ is not simple is important here either - I tried to use the fact that the multiplicity of $F$ at $P$ will be greater than 1 along with the fact $I(P,F\cap E)=m_P(F)\cdot m_P(E)$ when $F,E$ have no common tangents, but this hasn't worked.

FireGarden
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1 Answers1

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For $P=(0,0)$ and $F=Y^2-X^2-X^3, \; G=Y+X,\;H=Y-X$ we get $$I(P,F\cap (G+H))=2\quad \text{and} \quad I(P,F\cap G)=I(P,F\cap H))=3$$ and indeed $2\geq \min(3,3)$ is not true.

Georges Elencwajg
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