In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. 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. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov–Witten theory and the extension of intersection theory from schemes to stacks.
Topological intersection form
For a connected oriented manifold M {\displaystyle M} of dimension 2 n {\displaystyle 2n} the intersection form is defined on the n {\displaystyle n} -th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class [ M ] {\displaystyle [M]} in H 2 n ( M , ∂ M ) {\displaystyle H_{2n}(M,\partial M)} . Stated precisely, there is a bilinear form
λ M : H n ( M , ∂ M ) × H n ( M , ∂ M ) → Z {\displaystyle \lambda _{M}\colon H^{n}(M,\partial M)\times H^{n}(M,\partial M)\to \mathbf {Z} }
given by
λ M ( a , b ) = ⟨ a ⌣ b , [ M ] ⟩ ∈ Z {\displaystyle \lambda _{M}(a,b)=\langle a\smile b,[M]\rangle \in \mathbf {Z} }
with
λ M ( a , b ) = ( − 1 ) n λ M ( b , a ) ∈ Z . {\displaystyle \lambda _{M}(a,b)=(-1)^{n}\lambda _{M}(b,a)\in \mathbf {Z} .}
This is a symmetric form for n even (so 2n = 4k doubly even), in which case the signature of M is defined to be the signature of the form, and an alternating form for n odd (so 2n = 4k + 2 is singly even). These can be referred to uniformly as ε-symmetric forms, where ε = (−1)n = ±1 respectively for symmetric and skew-symmetric forms. It is possible in some circumstances to refine this form to an ε-quadratic form, though this requires additional data such as a framing of the tangent bundle. It is possible to drop the orientability condition and work with Z/2Z coefficients instead. These forms are important topological invariants. For example, a theorem of Michael Freedman states that simply connected compact 4-manifolds are (almost) determined by their intersection forms up to homeomorphism. By Poincaré duality, it turns out that there is a way to think of this geometrically. If possible, choose representative n-dimensional submanifolds A, B for the Poincaré duals of a and b. Then λM (a, b) is the oriented intersection number of A and B, which is well-defined because since dimensions of A and B sum to the total dimension of M they generically intersect at isolated points. This explains the terminology intersection form.
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.
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