nLab intersection theory

Idea

Intersection theory studies literally the intersection of pairs of sub-spaces inside an ambient space. Historically Bézout's theorem described the signed (integer-valued) number of points at which two algebraic curves meet transversally inside an ambient algebraic surface. Dually, under Poincaré duality, this integer is the evaluation of the cup product of the duals of the two fundamental classes of the curves on that of the ambient spaces, and hence the cup product on mid-degree cohomology classes is also called the intersection product and is one central aspect of intersection theory.

However, if the sub-spaces do not intersect sufficiently transversally, then their plain set-theoretic number of intersection points will not agree with the cohomological intersection product thus defined. Historically Serre's intersection formula was a first step to remedy this by replacing the plain tensor product of structure sheaves (which is what dually describes the intersection of the varieties) by a Tor, hence by its derived functor.

In the modern version of the theory (as indicated e.g. in the introduction of (Lurie-Spaces)) this is interpreted as saying that the intersection is to be taken in derived algebraic geometry (and the fundamental classes are to be taken to be virtual fundamental classes).

With intersection theory interpreted in (infinity,1)-topos theory this way then the nature of Bézout’s original statement should hold in full generality essentially without further assumptions and this would be the central statement of intersection theory:

the cup product of two virtual fundamental classes of mid-dimensional subspaces equals the fundamental class of their derived intersection.

Related concepts

• intersection product

• Gromov-Witten invariant,

• Riemann-Roch theorem,

• enumerative geometry,

• Bezout theorem,

• virtual fundamental class, Mumford class, homological mirror symmetry, Schubert calculus

References

Introductions and surveys include

• Wikipedia, Intersection theory

• William Fulton, Introduction to intersection theory in algebraic geometry, CBMS 54, AMS, 1996, second edition

• Akhil Mathew, Intersection theory on surfaces, 2013 (web)

• Günther Trautmann, Introduction to intersection theory (2007) $[$pdf$]$

The perspective in derived algebraic geometry was clearly articulated in the introduction of

• Jacob Lurie, Structured Spaces

See also

• David Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328, MR85j:14046

• Maxim Kontsevich, Intersection theory on the moduli spaces of curves and the matrix. Airy function, Comm.Math.Phys.,vol.147(1992),1-23, pdf; Intersection theory on the moduli spaces of curves, Functional.Anal.Appl.,25:2(1991) pdf