Intersection theory

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In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand the topological theory more quickly reached a definitive form.

[edit] Topological intersection form

For a connected oriented manifold M of dimension 2n the intersection form is defined on the n^th cohomology group (what is usually called the 'middle dimension') in the form of the cup product (in what follows, we can drop the orientability condition and work with coefficients). Stated precisely, there is a bilinear form

given by

This is a quadratic form for n even, and an alternating form for n odd, because of the graded-commutative nature of the cohomology ring. These forms are important topological invariants. For example, a theorem of Michael Freedman states that simply connected smooth 4-manifolds are determined by their intersection forms up to homeomorphism.

By Poincare duality, it turns out that there is a way to think of this geometrically. Choose representative n-dimensional submanifolds A, B for the Poincaré duals of a and b. Then Q_M(a, b) is the oriented intersection number of A and B, which is well-defined because of the dimensions of A and B. This explains the terminology intersection form.

[edit] Intersection theory in algebraic geometry

William Fulton in Intersection Theory (1984) writes

... if A and B are subvarieties of a non-singular variety X, the intersection product A.B should be an equivalence class of algebraic cycles closely related to the geometry of how A∩B, A and B are situated in X. Two extreme cases have been most familiar. If the intersection is proper, i.e. dim(A∩B) = dim A + dim B − dim X, then A.B is a linear combination of the irreducible components of A∩B, with coefficients the intersection multiplicities. At the other extreme, if A = B is a non-singular subvariety, the self-intersection formula says that A.B is represented by the top Chern class of the normal bundle of A in X.

To give a definition, in the general case, of the intersection multiplicity was the major concern of André Weil's 1946 book Foundations of Algebraic Geometry. Work in the 1920s of B. L. van der Waerden had already addressed the question; in the Italian school of algebraic geometry the ideas were well known, but foundational questions were not addressed in the same spirit.

Self-intersection is a key idea, for example in birational geometry. There on an algebraic surface S, blowing up creates from a point a curve C recognisable by its genus, which is 0, and its self-intersection number, which is −1. There is no paradox, since while C∩C is C as a set, C.C does not mean that set-theoretic intersection. Consider a line L in the projective plane: it has self-intersection number 1 since all other lines cross it once. A line on a non-singular quadric Q in P³ has self-intersection 0, since Q is also P¹×P¹ and a line P¹ can be moved off itself. The quadric Q projects to the plane by means of lines through a fixed point on it. In terms of intersection forms the plane has one of type x² while the quadric one of type XY. This happens on the addition of a negative term to x² − y², and then a change of basis.