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.
Then the intersection number of two closed curves on X has a simple definition in terms of an integral.
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
Then define fc by The definition is then expanded to arbitrary closed curves.
by 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.
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 (called the intersection multiplicity at x) is 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.
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.
First, even if the expected dimension of V ∩ W is zero, the actual intersection may be of a large dimension: for example 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, if V is a plane curve and W is one of its tangent lines.
The first problem requires the machinery of intersection theory, discussed above in detail, which replaces 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.
Serre's Tor formula states: let X be a regular variety, V and W two subvarieties of complementary dimension such that V ∩ W is zero-dimensional.
When V and W can be moved into a transverse position, this homological formula produces the expected answer.
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.
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
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: The intersection number of line bundles Li's is then defined by: where χ denotes the Euler characteristic.
One realization of intersection multiplicity is through the dimension of a certain quotient space of the power series ring
If the original equations are given in homogeneous form, these can be obtained by setting
define curves which intersect only once in the closure of an open set
are smooth and intersect transversally (i.e. have different tangent lines) at exactly some number
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.