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.
Best Answer
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.