GitHub

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.

The varieties we are interested in are smooth projective varieties over the complex numbers.

Note

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." The authors of Schubert2 are 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 is due to Dang Tuan Hiep. All this work, including ours, is inspired by the Maple package Schubert written by Sheldon Katz and Stein A. Strømme.

The starting point of the original Schubert package was the problem of enumerating twisted cubic curves on a general quintic hypersuface in $\mathbb P^4$, see [ES02]. We quote from that paper:

One way to approach enumerative problems is to find a suitable complete parameter space for the objects that one wants to count, and express the locus of objects satisfying given conditions as a certain zero-cycle on the parameter space. For this method to yield an explicit numerical answer, one needs in particular to be able to evaluate the degree of a given zerodimensional cycle class [(integration)]. This is possible in principle whenever the numerical intersection ring (cycles modulo numerical equivalence) of the parameter space is known, say in terms of generators and relations.

Note

Following the authors of Schubert, we work with cycles modulo numerical equivalence rather than rational equivalence. Nevertheless, abusing our notation, we refer to the resulting intersection rings as Chow rings. These rings are graded by the codimension of cycles.

As in Schubert, our approach is abstract in the sense that we do not work with explicit varieties 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 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 the Chow rings of abstract flag bundles see

Tutorials

We encourage you to take a look at the tutorials on intersection theory in OSCAR, which can be found here.

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.