Intersection number

Intersection number

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, and especially in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, 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
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

. One needs a definition of intersection number in order to state results like Bézout's theorem
Bézout's theorem
Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...

.

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
Intersection theory
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...

.

Definition for Riemann surfaces

Let X be a Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

. 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 ), we can associate a differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

with the pleasant property that integrals along c can be calculated by integrals over X:
, for every closed (1-)differential on X,

where is the wedge product of differentials, and is the hodge star. Then the intersection number of two closed curves, a and b, on X is then defined as
.

The have an intuitive definition as follows. They are a sort of dirac delta along the curve c, accomplished by taking the differential of a unit step function that drops from 1 to 0 across c. More formally, we begin by defining for a simple closed curve c on X, a function fc by letting be a small strip around c in the shape of an annulus. Name the left and right parts of as and . Then take a smaller sub-strip around c, , with left and right parts and . Then define fc by
.

The definition is then expanded to arbitrary closed curves. Every closed curve c on X is homologous to for some simple closed curves ci, that is,
, for every differential .

Define the by
.

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 divisor
Divisor (algebraic geometry)
In algebraic geometry, divisors are a generalization of codimension one subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors...

s 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:
• .
• for all i. (i.e., x is in the intersection of the hypersurfaces.)
• (i.e., the divisors are in general position.)
• The are nonsingular at x.

Then the intersection number at the point x is
,

where is the local ring of X at x, and the dimension is dimension as a k-vector space. It can be calculated as the localization
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...

, where is the maximal ideal of polynomials vanishing at x, and U is an open affine set containing x and containing none of the singularities of the fi.

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.,
and .

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 "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
Cup product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...

. 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 of the Poincaré duals of X and Y.

Intersection multiplicities for plane curves

There is a unique function assigning to each triplet (PQp) consisting of a pair of polynomials, P and Q, in K[xy] and a point p in K2 a number Ip(PQ) 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[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 Kx,y. By making a change of variables if necessary, we may assume that the point p is (0,0). Let P(xy) and Q(xy) 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 = (PQ) denote the ideal of Kx,y generated by P and Q. The intersection multiplicity is the dimension of Kxy/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
Degree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...

of P with respect to x is equal to the total degree of P, Ip(PQ) 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
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

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. Ip(PQ) = n.

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
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

.

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 divisor
Divisor (algebraic geometry)
In algebraic geometry, divisors are a generalization of codimension one subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors...

s of some specific kind, two representatives are intersected that are in general position
General position
In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...

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
Bézout's theorem
Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...

.

The intersection number arises in the study fixed point
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

s, which can be cleverly defined as intersections of function graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

s with a diagonal
Diagonal
A diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...

s. 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.