How to compute the intersection number at singularities?

Question

I'm studying algebraic curves and compact riemann surfaces. I read several related nice textbooks such as Kirwan, Griffiths and Miranda.

I met such a concrete example in Griffiths' book: on page 206, test problem 5, I have to compute the intersection number at $(0,0)$ of the following pairs of curves:

(A) $x^3-x^2+y^2=0$ and $y^2=x^3$. (B) $(y-x)^3=4\sqrt{2}xy$ and $y^2=x^3$.

I was at a loss. Because on [Miranda], the intersection number is defined for smooth curves using the order of meromorphic function on riemann surfaces. It can't deal with singularities. And [Kirwan] define the intersection number purely algebraically. It only deals with the existence and then prove the Bezout theorem. Griffiths' own book defines this notion using the idea of local "normaloztion", but I don't know how to use this definition to compute these examples.

Can any veteran give me some suggestions? Thank you very much :)

Answer 1

Here is a solution for (A).
The required intersection number $I_{(0,0)}=I_p$ only depends on the ideal generated by the equations of the curves, here that ideal is $(x^3-x^2+y^2, y^2-x^3)$.
On page 59 Kirwan gives rules that allow you to manipulate the two generators of the ideal until you are reduced to a trivial calculation. In case (A) you obtain :
$$I_p=I_p(x^3-x^2+y^2, y^2-x^3)= I_p(x^3-x^2+y^2-(y^2-x^3),y^2-x^3)= I_p(x^3-x^2+x^3,y^2-x^3)= I_p(x^2(2x-1),y^2-x^3)=I_p(x^2,y^2-x^3)+ I_p(2x-1,y^2-x^3)=I_p(x^2,y^2-x^3)+0=I_p(x^2,y^2-x^3+x^2\cdot x)=I_p(x^2,y^2)=4$$ I have used that $I_p(2x-1,y^2-x^3)=0$ and the reason for that is that the curve (actually an affine line) $2x-1=0$ does not pass through $p$: cf. Kirwan Theorem 3.18 (iii).
Beware that Griffiths's remark that the first curve in part (B) is obtained by rotating 45° the first curve in part (A) is true but completely irrelevant: rotating a curve may change its intersection with another fixed curve.

Answer 2

Fulton describes an algorithm for computing multiplicities (§3.3, p.37, example on p.40). I try to give some intuition for the algorithm here. Briefly, Fulton characterizes multiplicity with a list of seven property, of which (7) is perhaps the most important: $I_P(E\cap F) = I_P(E\cap(F+AE))$ for any polynomial $A$. We want to use this to simplify the curves, reducing the total degree when possible. If that doesn't work, reduce the degree of the "x-part", i.e., $E(x,0)$. (Or the y-part: $E(0,y)$.)

Kendig has extensive, well-motivated discussions and many examples for multiplicity calculations, including the case of singularities like $y^2=x^3$ (see p.51.) (I've been blogging about both these books.)