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Then the intersection numbers of H and L are given by H 5 =1P, H 4 L=2P, H 3 L 2 =4P, H 2 L 3 =4P, H 1 L 4 =2P, L 5 =1P. So we have (6 H −2 E ) 5 = (2 H +2 L ) 5 = 3264 P . Fulton & MacPherson gave a precise description of exactly what "general position" means (although their two propositions about this are not quite right, and are corrected ...
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.)
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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 ...
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]
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.
In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1.
In 1987 he moved to the University of Chicago. [1] He is, as of 2011, a professor at the University of Michigan. [2] As of 2024, Fulton had supervised the doctoral work of 24 students at Brown, Chicago, and Michigan. Fulton is known as the author or coauthor of a number of popular texts, including Algebraic Curves and Representation Theory.