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I'm reading Algebraic Curves by Fulton where the concept of intersection numbers is introduced on page 36. They give both a definition $$I(P,F\cap G) = \mathcal{O}_P(\mathbb{A^2})/(F,G)$$ And a characterisation using 7 properties. The 5th property of those is: $$I(P,F\cap G) \geq m_p(F)m_p(G)$$ Where $m_P(F)$ is the multiplicity of $P$ on $F$. They also discuss when this inequality is an equality but that is not part of my question. The proof they give is the following:

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Question: Where do they use the fact that $m$ and $n$ are the multiplicities involved? What part of the proof would no longer be true if we changed $m$ or $n$?

Jens Renders
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1 Answers1

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I think it's in the definition of $\psi$. We don't want $\psi(\overline A, \overline B) = \overline{AF + BG}$ to depend on the representative for $\overline A = A + I^n$ or $\overline B = B + I^m$. So if $N \in I^n$ and $M \in I^m$ then we want

$$NF + MG \in I^{m + n}.$$

This works because $m_P(F) = m$ (where $P = (0,0)$) says exactly that $F$ consists of terms of degree $\ge m$ which is exactly the statement $F \in I^m$. Likewise, $G \in I^n$. Thus $NF + MG \in I^{m + n}$.

Trevor Gunn
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  • Thanks! Troughout this whole book I've been checking if the maps that are used are well defined, but this time I just overlooked it! Nice explaination too! – Jens Renders Jul 31 '17 at 10:21