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Steiner claimed that the number of conics tangent to 5 given conics in general position is 7776 = 6 5, but later realized this was wrong. [2] The correct number 3264 was found in about 1859 by Ernest de Jonquières who did not publish because of Steiner's reputation, and by Chasles using his theory of characteristics, [3] and by Berner in 1865.
One says that “the affine plane does not have a good intersection theory”, and intersection theory on non-projective varieties is much more difficult. A line on a P 1 × P 1 (which can also be interpreted as the non-singular quadric Q in P 3) has self-intersection 0, since a line can be moved off itself. (It is a ruled surface.)
The correct number 3264 was found by Berner in 1865 and by Ernest de Jonquieres around 1859 and by Chasles in 1864 using his theory of characteristics. However these results, like many others in classical intersection theory, do not seem to have been given complete proofs until the work of Fulton and Macpherson in about 1978. Dirichlet's principle.
Fulton and MacPherson extended the Chow ring to singular varieties by defining the "operational Chow ring" and more generally a bivariant theory associated to any morphism of schemes. [13] A bivariant theory is a pair of covariant and contravariant functors that assign to a map a group and a ring respectively.
The study of moduli spaces of curves, maps and other geometric objects, sometimes via the theory of quantum cohomology. The study of quantum cohomology, Gromov–Witten invariants and mirror symmetry gave a significant progress in Clemens conjecture. Enumerative geometry is very closely tied to intersection theory. [1]
In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is , the fiber product of the closed immersions ,. It is denoted by X ∩ Y {\displaystyle X\cap Y} . Locally, W is given as Spec R {\displaystyle \operatorname {Spec} R} for some ring R and X , Y as Spec ( R / I ) , Spec ( R / J ...
Eisenbud, David; Harris, Joe (2016), 3264 and All That: A Second Course in Algebraic Geometry, Cambridge University Press, ISBN 978-1107602724; Kleiman, Steven L. (1974), "The transversality of a general translate", Compositio Mathematica, 28: 287– 297, MR 0360616
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.