In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem.
Contents
- Definition for Riemann surfaces
- Definition for algebraic varieties
- Further definitions
- Intersection multiplicities for plane curves
- Example
- Self intersections
- Applications
- References
The intersection number is obvious in certain cases, such as the intersection of x- and y-axes which should be one. The complexity enters when calculating intersections at points of tangency and intersections along positive dimensional sets. For example, if a plane is tangent to a surface along a line, the intersection number along the line should be at least two. These questions are discussed systematically in intersection theory.
Definition for Riemann surfaces
Let X be a Riemann surface. Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function
where
The
The definition is then expanded to arbitrary closed curves. Every closed curve c on X is homologous to
Define the
Definition for algebraic varieties
The usual constructive definition in the case of algebraic varieties proceeds in steps. The definition given below is for the intersection number of divisors on a nonsingular variety X.
1. The only intersection number that can be calculated directly from the definition is the intersection of hypersurfaces (subvarieties of X of codimension one) that are in general position at x. Specifically, assume we have a nonsingular variety X, and n hypersurfaces Z1, ..., Zn which have local equations f1, ..., fn near x for polynomials fi(t1, ..., tn), such that the following hold:
Then the intersection number at the point x is
where
2. The intersection number of hypersurfaces in general position is then defined as the sum of the intersection numbers at each point of intersection.
3. Extend the definition to effective divisors by linearity, i.e.,
4. Extend the definition to arbitrary divisors in general position by noticing every divisor has a unique expression as D = P - N for some effective divisors P and N. So let Di = Pi - Ni, and use rules of the form
to transform the intersection.
5. The intersection number of arbitrary divisors is then defined using a "Chow's moving lemma" that guarantees we can find linearly equivalent divisors that are in general position, which we can then intersect.
Note that the definition of the intersection number does not depend on the order of the divisors.
Further definitions
The definition can be vastly generalized, for example to intersections along subvarieties instead of just at points, or to arbitrary complete varieties.
In algebraic topology, the intersection number appears as the Poincaré dual of the cup product. Specifically, if two manifolds, X and Y, intersect transversely in a manifold M, the homology class of the intersection is the Poincaré dual of the cup product
Intersection multiplicities for plane curves
There is a unique function assigning to each triplet
-
I p ( P , Q ) = I p ( Q , P ) -
I p ( P , Q ) = ∞ if and only ifP andQ have a common factor that is zero atp -
I p ( P , Q ) = 0 if and only if one ofP ( p ) orQ ( p ) is non-zero (i.e. the pointp is off one of the curves) -
I p ( x , y ) = 1 wherep = ( 0 , 0 ) -
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 anyR ∈ 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
Another realization of intersection multiplicity comes from the resultant of the two polynomials
Intersection multiplicity can also be realised as the number of distinct intersections that exist if the curves are perturbed slightly. More specifically, if
Example
Consider the intersection of the x-axis with the parabola
Then
and
so
Thus, the intersection degree is two; it is an ordinary tangency.
Self-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.
Applications
The intersection number is partly motivated by the desire to define intersection to satisfy Bézout's theorem.
The intersection number arises in the study of fixed points, which can be cleverly defined as intersections of function graphs with a diagonals. Calculating the intersection numbers at the fixed points counts the fixed points with multiplicity, and leads to the Lefschetz fixed point theorem in quantitative form.