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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 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]
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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.
encodes all the intersection indices as its coefficients. Witten's conjecture states that the partition function Z = exp F is a τ-function for the KdV hierarchy , in other words it satisfies a certain series of partial differential equations corresponding to the basis { L − 1 , L 0 , L 1 , … } {\displaystyle \{L_{-1},L_{0},L_{1},\ldots ...
Intersection homology was originally defined on suitable spaces with a stratification, though the groups often turn out to be independent of the choice of stratification. There are many different definitions of stratified spaces. A convenient one for intersection homology is an n-dimensional topological pseudomanifold.
For example, the expected dimension of intersection of and is , the intersection of and has expected dimension , and so on. The definition of a Schubert variety states that the first value of j {\displaystyle j} with dim ( V j ∩ w ) ≥ i {\displaystyle \dim(V_{j}\cap w)\geq i} is generically smaller than the expected value n − k + i ...