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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.)
Pages in category "Intersection theory" The following 13 pages are in this category, out of 13 total. This list may not reflect recent changes. ...
A door is an example of a complex feature that is seemingly trivial to implement correctly. In the original description of the analogy, Liz England justifies and explains the job requirements of a designer and how complex the job actually is compared to how the requirements are initially posed (making a door).
That is, a scheme-theoretic multiplicity of an intersection may differ from an intersection-theoretic multiplicity, the latter given by Serre's Tor formula. Solving this disparity is one of the starting points for derived algebraic geometry, which aims to introduce the notion of derived intersection.
The second potential problem is that even if the intersection is zero-dimensional, it may be non-transverse, for example, if V is a plane curve and W is one of its tangent lines. The first problem requires the machinery of intersection theory, discussed above in detail, which replaces V and W by more convenient subvarieties using the moving lemma.
"K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems". arXiv: math/9908097. Lowrey, Parker; Schürg, Timo (2012-08-30). "Grothendieck-Riemann-Roch for derived schemes". arXiv: 1208.6325 . Vakil, Math 245A Topics in algebraic geometry: Introduction to intersection theory in algebraic geometry
The intersection points are: (−0.8587, 0.7374, −0.6332), (0.8587, 0.7374, 0.6332). A line–sphere intersection is a simple special case. Like the case of a line and a plane, the intersection of a curve and a surface in general position consists of discrete points, but a curve may be partly or totally contained in a surface.
To be precise, they develop the intersection theory by a way of solving the problems of residual intersections (namely, by the use of the Segre class of a normal cone to an intersection.) A generalization to a situation where the assumption on regular embedding is weakened is due to Kleiman (1981).