I am wondering about the geometric significance of the intersection multiplicity of two curves as defined in Hartshorne 5.4 (The length of $O_p/(f,g)$ is the intersection multiplicity of $Z(f)$ and $Z(g)$ at $P$). Can someone provide a reference or quick description to address what this means geometrically? From part c) it seems like it has something to do with the number of repeated roots of an equation, but I'm not sure.
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The modern definition is a distillation of hundreds of years of calculations of intersections of curves.
A good intuitive description is that the intersection number $I_p(f,g)$ at $p$ of two curves $Z(f), Z(g)$ lying on $\mathbb A^2$ is the number of actual points of intersection near $p$ of the two curves obtained by slightly deforming the equations of the first curve to $f'$ .
For example if your curves are given by $f=y$ and $g=y^2-x^3$ they only have one point of intersection $O=(0,0)$ .
But if you deform $f$ to $f'=y-\epsilon$ for some small $\epsilon \gt0$ , the curves $Z(f')$ and $Z(g)$ will intersect in the three points $(\epsilon^{2/3},\epsilon),(\omega \epsilon^{2/3},\epsilon),(\omega^2 \epsilon^{2/3},\epsilon)$ [where $\omega$ is a primitive third root of unity] .
This illustrates that the length of $\mathcal O_{\mathbb A^2,O}/(y,y^2-x^3)$ is indeed $3$.
Georges Elencwajg
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Just to add, because I think you mean Hartshorne Chapter 1, section 5, exercise 4, NOT chapter 5 of Hartshorne. The intersection multiplicity, as it turns out, is the unique way of pairing curves against each other which satisfy some natural conditions, the most intuitive of which is that if the two curves meet transversely (don't have any agreement of tangency) you should count that intersection with multiplicity $1$. So, somehow, this definition was forced upon us all along. it should also be probably noted that if you're working over $\bar{k}$, you can replace this length with dimension. – Alex Youcis Oct 26 '14 at 00:32