Intersection Number - Intersection Multiplicities For Plane Curves

Intersection Multiplicities For Plane Curves

There is a unique function assigning to each triplet (P, Q, p) consisting of a pair of polynomials, P and Q, in K and a point p in K2 a number I_p(P, Q) called the intersection multiplicity of P and Q at p that satisfies the following properties:

1. is infinite if and only if P and Q have a common factor that is zero at p.
2. is zero if and only if one of P(p) or Q(p) is non-zero (i.e. the point p is off one of the curves).
3. where the point p is at (x, y).
4. for any R in K

Although these properties completely characterize intersection multiplicity, in practice it is realised in several different ways.

One realization of intersection multiplicity is through the dimension of a certain quotient space of the power series ring K]. By making a change of variables if necessary, we may assume that the point p is (0,0). Let P(x, y) and Q(x, y) be the polynomials defining the algebraic curves we are interested in. If the original equations are given in homogeneous form, these can be obtained by setting z = 1. Let I = (P, Q) denote the ideal of K] generated by P and Q. The intersection multiplicity is the dimension of K]/I as a vector space over K.

Another realization of intersection multiplicity comes from the resultant of the two polynomials P and Q. In coordinates where p is (0,0), the curves have no other intersections with y = 0, and the degree of P with respect to x is equal to the total degree of P, I_p(P, Q) can be defined as the highest power of y that divides the resultant of P and Q (with P and Q seen as polynomials over K).

Intersection multiplicity can also be realised as the number of distinct intersections that exist if the curves are perturbed slightly. More specifically, if P and Q define curves which intersect only once in the closure of an open set U, then for a dense set of (ε,δ) in K2, P − ε and Q − δ are smooth and intersect transversally (i.e. have different tangent lines) at exactly some number n points in U. I_p(P, Q) = n.