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 such as Bezout's theorem.

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 of an 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.

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" [x,y] and a point p in $K^2$ a number $I\_p(P,Q)$ called the "intersection multiplicity" of P and Q at p that satisfies the following properties:

# $I\_p(P,Q)\; =\; I\_p(Q,P)$.
# $I\_p(P,Q)$ is infinite if and only if P and Q have a common factor that is zero at p.
# $I\_p(P,Q)$ is zero if and only if one of P(p) or Q(p) is non-zero (i.e. the point p is not on one of the curves).
# $I\_p(x,y)\; =\; 1$ where the point "p" is at "(x, y)".
# $I\_p(P,Q\_1*Q\_2)\; =\; I\_p(P,Q\_1)\; +\; I\_p(P,Q\_2)$
# $I\_p(P\; +\; Q*R,Q)\; =\; I\_p(P,Q)$ for any $R$ in "K" [x,y]

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""x","y". 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""x","y" generated by "P" and "Q". The intersection multiplicity is the dimension of 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" [x] ).

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 $K^2$, P−ε and Q−δ are smooth and intersect transversally (i.e. have different tangent lines) at exactly some number n points in U. $I\_p(P,Q)\; =\; n$.

Example

Consider the intersection of the x-axis with the parabola

:"y" = "x"².

Then

:"P" = "y",

and

:"Q" = "y" − "x"²,

so

: $I\_p(P,Q)\; =\; I\_p(y,y\; -\; x^2)\; =\; I\_p(y,x^2)\; =\; I\_p(y,x)\; +\; I\_p(y,x)\; =\; 1\; +\; 1\; =\; 2$.

Thus, the intersection degree is two; it is an ordinary tangency.

elf-intersections

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 divisors 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.

Reference

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* Appendix A.