.
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.
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 (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.,
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 in which the divisors appear in the computation of this number.
![{\displaystyle k[U]_{{\mathfrak {m}}_{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50e9c2e3b6b55dec2ce2bab50634878fa8b97941)
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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.
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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. 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
; in particular, and commute.
(this is nontrivial and follows from a .)
dévissage argument
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Self-intersections[edit]
Some of the most interesting intersection numbers to compute are self-intersection numbers. This means that a divisor is moved to another equivalent divisor in general position with respect to the first, and the two are intersected. 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.
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