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Concept
Intersection number
Résumé
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 the x- and y-axes in a plane, which should be one. The complexity enters when calculating intersections at points of tangency, and intersections which are not just points, but have higher dimension. 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.
Differential forms on a Riemann surface#Duality between 1-forms and closed curves
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, the Poincaré dual of c, 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 .
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