Intersection theory

On the other hand, the topological theory more quickly reached a definitive form.

For example, a theorem of Michael Freedman states that simply connected compact 4-manifolds are (almost) determined by their intersection forms up to homeomorphism.

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.

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.

The guiding principle in the definition of intersection multiplicities of cycles is continuity in a certain sense.

The first fully satisfactory definition of intersection multiplicities was given by Serre: Let the ambient variety X be smooth (or all local rings regular).

Remarks: The Chow ring is the group of algebraic cycles modulo rational equivalence together with the following commutative intersection product: whenever V and W meet properly, where

In the plane, this just means translating the curve C in some direction, but in general one talks about taking a curve C′ that is linearly equivalent to C, and counting the intersection C · C′, thus obtaining an intersection number, denoted C · C. Note that unlike for distinct curves C and D, the actual points of intersection are not defined, because they depend on a choice of C′, but the “self intersection points of C′′ can be interpreted as k generic points on C, where k = C · C. More properly, the self-intersection point of C is the generic point of C, taken with multiplicity C · C. Alternatively, one can “solve” (or motivate) this problem algebraically by dualizing, and looking at the class of [C] ∪ [C] – this both gives a number, and raises the question of a geometric interpretation.

Note that passing to cohomology classes is analogous to replacing a curve by a linear system.

Note that on the affine plane, one might push off L to a parallel line, so (thinking geometrically) the number of intersection points depends on the choice of push-off.

A key example of self-intersection numbers is the exceptional curve of a blow-up, which is a central operation in birational geometry.

Note that as a corollary, P2 and P1 × P1 are minimal surfaces (they are not blow-ups), since they do not have any curves with negative self-intersection.

In fact, Castelnuovo’s contraction theorem states the converse: every (−1)-curve is the exceptional curve of some blow-up (it can be “blown down”).