Intersection number

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. 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. 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 c : S 1 → X {displaystyle c:S^{1} o X} ), we can associate a differential form η c {displaystyle eta _{c}} of compact support with the property that integrals along c can be calculated by integrals over X: where ∧ {displaystyle wedge } is the wedge product of differentials, and ∗ {displaystyle *} is the Hodge star. Then the intersection number of two closed curves, a and b, on X is defined as The η c {displaystyle eta _{c}} 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 Ω {displaystyle Omega } be a small strip around c in the shape of an annulus. Name the left and right parts of Ω ∖ c {displaystyle Omega setminus c} as Ω + {displaystyle Omega ^{+}} and Ω − {displaystyle Omega ^{-}} . Then take a smaller sub-strip around c, Ω 0 {displaystyle Omega _{0}} , with left and right parts Ω 0 − {displaystyle Omega _{0}^{-}} and Ω 0 + {displaystyle Omega _{0}^{+}} . Then define fc by The definition is then expanded to arbitrary closed curves. Every closed curve c on X is homologous to ∑ i = 1 N k i c i {displaystyle sum _{i=1}^{N}k_{i}c_{i}} for some simple closed curves ci, that is, Define the η c {displaystyle eta _{c}} 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. 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: