Introduction
In this chapter, we
- introduce OSCAR tools which support computations in intersection theory, and
- give examples which illustrate how intersection theory is used to
solve problems from enumerative geometry.
Making use of OSCAR, a first version of what we present here was written by Jeiao Song as a julia package.
Throughout the chapter, the varieties we consider are smooth projective varieties over the complex numbers.
The Chow ring of a variety X is the group of cycles of X modulo an equivalence relation, together with the intersection pairing which defines the multiplication of the ring. Here, in contrast to most textbooks, we consider numerical equivalence classes of cycles rather than rational equivalence classes.
Our approach is abstract in the sense that we do not work with concrete varieties; that is, our varieties are not given by equations. Instead, we represent a variety by specifying its dimension together with its Chow ring and, possibly, further data. We refer to such such a collection of data as an abstract variety, and to results obtained from manipulating the data as results which apply to all (smooth projective complex) varieties sharing the data.
Of particular interest is the tangent bundle of a variety (recall that the Todd class of the tangent bundle enters the Hirzebruch-Riemann-Roch formula). As with any other vector bundle, the tangent bundle is represented as a collection of data referred to as an abstract vector bundle. The main data here is the Chern character polynomial of the vector bundle.
In the same spirit, we introduce abstract variety maps. Their key ingredient is the pullback morphism between the Chow rings.
General textbooks offering details on theory and algorithms include:
For computations in the Chow rings of abstract flag bundles see
- [GSS22].
Contact
Please direct questions about this part of OSCAR to the following people:
You can ask questions in the OSCAR Slack.
Alternatively, you can raise an issue on github.