Intersection number

In mathematics, the concept of intersection number arose in algebraic geometry, where two curves intersecting at a point may be considered to 'meet twice' if they are tangent there. In the sense that 'multiple intersections' are limiting cases of n-fold intersections at n points which come into coincidence, one needs a definition of intersection number in order to state theorems about counting intersections in a precise way.

The requirements of the general theory are to handle intersections in any dimensions, and in algebraic topology as well. For example, theorems about fixed points are about intersections of function graphs with diagonals; one wishes to count fixed points with multiplicity in order to have the Lefschetz fixed point theorem in quantitative form.

More general intersections will have higher-dimensional subsets or subvarieties in common, and one wants also to be able to talk of the intersection multiplicity of such an intersection, or irreducible component of it. For example if a plane is tangent to a surface along a line, that line should be counted with multiplicity two, at least. These questions are discussed systematically in intersection theory.

Some of the most interesting intersection numbers to compute are self-intersection numbers. This should not be taken in a naive sense. What is meant is that, in an equivalence class of some specific kind, two representatives are intersected that are in general position with respect to each other. In this way, self-intersection numbers can become well-defined, and even negative.

Intersection multiplicities for plane curves

In order to make Bézout's theorem definite, we need a precise definition of intersection multiplicity at a point. Here is one way to do it.

By making a change of variables if necessary, we may assume that the point we are interested in is (0:0:1) (that is, (0,0) in affine coordinates). Let f(x,y) and g(x,y) be the polynomials defining X and Y respectively; if the original equations are given in homogeneous form, these can be obtained by setting z=1. Let R = K[[x,y]] be the power series ring in x and y, and let I=(f,g) denote the ideal of R generated by f and g.

Then the intersection multiplicity is the dimension of R/I as a vector space over K. This is infinite if and only if the curves defined by f and g have a one-dimensional component in common passing through (0:0:1). If the two curves have no common component, the intersection multiplicity is finite at every point, and the assertion of Bézout's theorem makes sense.

Example: Let X be the x-axis and Y the parabola y = x². Then f=y, and g=y-x², so I=(y,y-x²) = (y,x²), which consists of all power series in which the coefficients of 1 and x are trivial. Thus, the intersection degree is two; it is an ordinary tangency.