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. This package was "heavily inspired by the Macaulay2 package Schubert2 and the Sage library Chow. Some functionalities from [the Sage library] Schubert3 are also implemented."
Schubert2 was written by Daniel R. Grayson, Michael E. Stillman, Stein A. Strømme, David Eisenbud, and Charley Crissman while Chow is due to Manfred Lehn and Christoph Sorger. Schubert3 as well as the Singular library schubert.lib were written by Dang Tuan Hiep. The basis for all this work, including ours, is the Maple package Schubert written by Sheldon Katz and Stein A. Strømme.
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.