Generalized notion of counting curve intersections
This article is about algebraic geometry. For the concept in graph theory, see Intersection number (graph theory).
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.
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