Intersection number

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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.

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[edit]

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 ), we can associate a differential form of compact support with the 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 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[edit]

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:

  • .
  • 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 (called the intersection multiplicity at 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 , 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 "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.

Serre's Tor formula[edit]

Let V and W be two subvarieties of a nonsingular projective variety X such that dim(V)+dim(W)=dim(X). Then we expect the intersection VW to be a finite set of points. If we try to count them, two kinds of problems may arise. First, even if the expected dimension of VW is zero, the actual intersection may be of a large dimension. For example, we could try to find the self-intersection number of a projective line in a projective plane. The second potential problem is that even if the intersection is zero-dimensional, it may be non-transverse. For example, V can be a tangent line to a plane curve W.

The first problem requires the machinery of intersection theory, discussed above in detail. The essential idea is to replace V and W by more convenient subvarieties using the moving lemma. On the other hand, the second problem can be solved directly, without moving V or W. In 1965 Jean-Pierre Serre described how to find the multiplicity of each intersection point by methods of commutative algebra and homological algebra.[1] This connection between a geometric notion of intersection and a homological notion of a derived tensor product has been influential and led in particular to several homological conjectures in commutative algebra.

The Serre's Tor formula is the following result. Let X be a regular variety, V and W two subvarieties of complementary dimension such that VW is zero-dimensional. For any point xVW, let A be the local ring of x. The structure sheaves of V and W at x correspond to ideals I, JA. Then the multiplicity of VW at the point x is

where length is the length of a module over a local ring, and Tor is the Tor functor. When V and W can be moved into a transverse position, this homological formula produces the expected answer. So, for instance, if V and W meet transversely at x, the multiplicity is 1. If V is a tangent line at a point x to a parabola W in a plane at a point x, then the multiplicity at x is 2.

If both V and W are locally cut out by regular sequences, for example if they are nonsingular, then in the formula above all higher Tor's vanish, hence the multiplicity is positive. The positivity in the arbitrary case is one of Serre's multiplicity conjectures.

Further definitions[edit]

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 of the Poincaré duals of X and Y.

Snapper–Kleiman definition of intersection number[edit]

There is an approach to intersection number, introduced by Snapper in 1959-60 and developed later by Cartier and Kleiman, that defines an intersection number as an Euler characteristic.

Let X be a scheme over a scheme S, Pic(X) the Picard group of X and G the Grothendieck group of the category of coherent sheaves on X whose support is proper over an Artinian subscheme of S.

For each L in Pic(X), define the endomorphism c1(L) of G (called the first Chern class of L) by

It is additive on G since tensoring with a line bundle is exact. One also has:

  • ; in particular, and commute.
  • (this is nontrivial and follows from a dévissage argument.)

The intersection number

of line bundles Li's is then defined by:

where χ denotes the Euler characteristic. Alternatively, one has by induction:

Each time F is fixed, is a symmetric functional in Li's.

If Li = OX(Di) for some Cartier divisors Di's, then we will write for the intersection number.

Let be a morphism of S-schemes, line bundles on X and F in G with . Then

.[2]

Intersection multiplicities for plane curves[edit]

There is a unique function assigning to each triplet consisting of a pair of projective curves, and , in and a point , a number called the intersection multiplicity of and at that satisfies the following properties:

  1. if and only if and have a common factor that is zero at
  2. if and only if one of or is non-zero (i.e. the point is off one of the curves)
  3. where
  4. for any

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 . By making a change of variables if necessary, we may assume that . Let and 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 . Let denote the ideal of generated by and . The intersection multiplicity is the dimension of as a vector space over .

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

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

Example[edit]

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[edit]

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[edit]

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.

Notes[edit]

  1. ^ Serre, Jean-Pierre (1965). Algèbre locale, multiplicités. Lecture Notes in Mathematics. 11. Springer-Verlag. pp. x+160.
  2. ^ Kollár 1996, Ch VI. Proposition 2.11

References[edit]