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Hartshorne defines the intersection multiplicity of a point in the intersection of two plane algebraic curves $f(x, y) = 0$, $g(x, y) = 0$ in $\mathbb{A}^{2}$ to be the length of the $\mathcal{O}_{P}$-module $\mathcal{O}_{P}/(f, g)$.

I have seen in other definitions that $\dim_{k}\mathcal{O}_{P}/(f, g)$ is used instead of the length, which I guess is the same in most cases.

Question: Does the above definition coincide with our geometric intuition of intersection multiplicity? I have seen very few related questions to this, all of which have unsatisfactory answers in regards to my question.

I have done some examples, such as $f(x, y) = y - x^{2}, g(x, y) = y - 2x - 1$ (the tangent line at $(1, 1)$). Then our desired ring is $\mathcal{O}_{(1, 1)}/(f, g) \cong k[x,y]_{(x-1, y-1)}/(y - x^{2}, y - 2x + 1) = k[x]_{(x-1)}/(x-1)^{2}$, which has $1, x$ as basis over $k$ and so has dimension $2$. So it works! But I have still no geometric insight from doing such an example.

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    $\begingroup$ Have you looked at the treatment of intersection multiplicity in section 3.3 of Fulton's curves book? He lists 7 axioms that intersection multiplicity should satisfy, and then shows that these axioms uniquely determine the intersection multiplicity, and that the definition (with dimension instead of length) satisfies them. $\endgroup$ – André 3000 Jul 16 '17 at 0:04
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    $\begingroup$ In terms of geometric intuition, you considered the ideal corresponding to the intersection of the parabola and its tangent line $(y - x^2, y - 2x - 1)$ and made a change of generators to get $(y - x^2, (x-1)^2)$. Since the vertical line $x=1$ intersects the parabola transversely, this square factor means that the intersection of the parabola and its tangent line is “worth” $2$ transverse intersections, i.e. the intersection multiplicity is $2$. $\endgroup$ – André 3000 Jul 16 '17 at 0:04
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    $\begingroup$ You ask "does above definition coincide with our geometric intuition of intersection multiplicity?" and the anwer is of course: why else would we define it like that! $\endgroup$ – Mariano Suárez-Álvarez Jul 16 '17 at 0:19
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    $\begingroup$ You could browse Fulton's book on curves or Reid's, which go into great length explaining this. Hartshorne's not the best source for geometric intuitions. $\endgroup$ – Mariano Suárez-Álvarez Jul 16 '17 at 0:20
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    $\begingroup$ @Quasicoherent Thank you for the link. I see now that the intersection multiplicity only depends on the ideal $(f, g)$, and, in the case of a tangent line, the change of coordinates should always get us a power of a linear factor. I guess that modding out by $(f, g)$ is doing something like ridding the higher-order terms so that we can "count" this power? I will think about that. $\endgroup$ – Fred Jul 16 '17 at 0:22
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$\text{}$1. The way I remember is that the intersection theory of plane curves is uniquely determined by being additive, stable under linear equivalence, and for two smooth curves intersecting transversally, it is the number of intersection points. See Chapter V, 1.1 and 1.4 of Hartshorne for the connection with the local multiplicity.

$\text{}$2. Fulton works for plane curves, whereas basically the same calculations work for nonsingular surfaces more generally. Finite length or dimension of a local ring over a field are equal, e.g. use the filtration by powers of the maximal ideal, with$$m^i/m^{i+1}$$a finite sequence of finite-dimensional vector spaces.

We want$$C_1 . C_2$$for curves of a surface to be a bilinear pairing under linear equivalence. The local definition is the thing that works. When you say intuition, you mean something you can understand without doing technical definitions. The definition as length of the local quotient$$\mathcal{O}_P/(f_1, f_2)$$depends only on $C_1$, $C_2$, whereas the intersection number equals$$(f_1 + \text{bit}).(f_2 + \text{bit})$$for $C_1$, $C_2$ moved by some tiny bit; you can arrange for the new $C_1'$ and $C_2'$ to meet transversally in a number of points equal to$$\text{dim}\,\mathcal{O}_P/(f_1, f_2),$$but then you have to prove that it is independent of the choice of $C_1'$ and $C_2'$—this is a dynamic definition.

You can get quite a long way just assuming that this dynamic definition works, but it is not so good for theoretical purposes, and to relate it to coherent cohomology.

As well as Fulton, there is quite a detailed discussion in Chapters 3 and 4 of Shafarevich.

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