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In the appendix of Algebaric Geometry by Hartshorne, he shows us that how Serre defines the intersection number in a more general case:$$i(X,Y;Z)=\sum(-1)^i\cdot\bigl(\operatorname{length} \operatorname{Tor}^A_i(A/a,A/b)\bigr),$$

where $X,Y$ intersect properly, $Z$ is an irreducible component, $A$ is the local ring of generic point of $Z$, and $a$ and $b$ are the ideals of $X$ and $Y$ in $A$. However, when computing a detailed example, such as when $X$ is a projective variety in $\mathbb P^n$ and $Y$ is a complete intersection defined by $f_1,...,f_r$, I find it difficult to write down what the intersection number is. Sincerely hope for a detailed computing method.

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I think it is difficult to compute the multiplicity just by looking "qualitatively" at the equations.

For instance, let $C \subset \mathbb{P}^3$ be a smooth curve over the field of complex numbers.

  • if $H \subset \mathbb{P}^3$ be a generic hyperplane. Then $H \cap C$ has intersection multiplicity $1$ at all points where it meets $C$.

  • let $c \in C$ be a generic point and let $H \subset \mathbb{P}^3$ be a generic hyperplane containing $T_{C,c}$. Then $H \cap C$ has multiplicity $2$ at $c$ and $1$ at the other points where it meets $C$ (the last part of this statement is non-trivial, it uses the Reflexivity Theorem in projective duality).

Of course one can distinguish easily between the first two cases because either $H$ is zero or not in $ \mathcal{M}_c/\mathcal{M}_c^2$ (where $\mathcal{M}_c$ is the maximal ideal of the point $c$).

On the other hand things become more ineteresting if you take $c_0 \in C$ such that $C$ has a flex point at $c_0$. Then, a general hyperplane containing $T_{C,c_0}$ has intersection multiplicity at least $3$ with $C$ at $c_0$.

Again things can be seen locally using higher differential spaces, but it becomes more and more tricky to explain what is going on. And here we have been dealing only with smooth space curves and linear spaces in $\mathbb{P}^3$. One can easily imagine that these intersection multiplicities are "very "hard" to compute in full generality.

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Let $X$ be a smooth projective variety, $V\subseteq X$ be any irreducible subvariety and $D\subseteq X$ be a prime Cartier divisor. Assume that $V$ and $D$ meet properly in $X$. Let $Z$ be an irreducible component of the intersection $V\cap D$. Set $A=\mathcal{O}_{X}(U)$, where $U$ is an affine open subset of X meeting $Z$ and such that $D$ is defined in $U$ by a local equation. Denote by $\mathfrak p\subseteq A$ the prime ideal corresponding to $V$ and by $a\in A_\mathfrak q$ the local equation of $D$. Then the intersection multiplicity of $V$ and $D$ along $Z$ is $$\mu_Z(V,D)=\textit{component at } \mathfrak q \textit{ of } (A/\mathfrak p)\cdot (a):=\ell_{A_\mathfrak q}\big(A/(\mathfrak pA_\mathfrak q+ aA_\mathfrak q)\big),$$ where $\mathfrak q$ is the prime corresponding to $Z$. It easy to check that the multiplicity defined above coincides with the usual one (use Koszul complexes). If we take a locally complete intersection irreducible subvariety $W\subseteq X$ in place of $D$, provided that the intersection with $V$ is proper, we can define the intersection multiplicity of $V$ and $W$ at an irreducible component $Z$ as $$\mu_Z(V,W)=(A_\mathfrak q/\mathfrak pA_\mathfrak q)\cdot (a_1)\cdots (a_k),$$ where the $a_{i}$'s are the local equations defining $W$ in some affine open subset $U\subseteq X$ meeting $Z$. Also in this case, one can prove that the intersection above coincides with the one defined by means of the Serre-Tor formula.

I sketched a construction that you can find in Thomas Geisser's paper Motivic Cohomology, K-Theory and Topological Cyclic Homology.

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