Introduction

A variety in this chapter is a smooth projective variety over the complex numbers. We will

• present OSCAR tools which support computations in the intersection theory of varieties, and
• give examples which illustrate how intersection theory is used to solve problems from enumerative geometry.

The starting point for developing the Schubert package was the problem of enumerating twisted cubic curves on a general quintic hypersurface 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. 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.

There are several adequate equivalence relations on algebraic cycles leading to useful cycle theories. The most common relation to work with is rational equivalence which yields intersection rings called Chow rings. In contrast to this usual practice, we follow the authors of Schubert and consider numerical equivalence, allowing rational coefficients of cycle classes. That is, if $N^\ast(X)$ is the numerical intersection ring of a variety $X$, we consider the ring

\[N^\ast(X)_{\mathbb Q} = N^\ast(X) \otimes_{\mathbb Z} {\mathbb Q},\]

which is graded by the codimension of cycles. In this chapter, the name Chow ring always refers to a ring of this type.

With respect to computations, again following Schubert, our approach is abstract in the sense that we do not work with explicit varieties given by equations. Instead, we specify the dimension of the variety 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 (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 of the vector bundle.

In the same spirit, we introduce abstract variety maps. Their key data is the pullback morphism between the respective Chow rings.

Some comments on this set-up are in order:

Let $X$ be a variety. Write $K^{\circ}(X)$ for the Grothendieck group of vector bundles on $X$. That is, $K^{\circ} (X)$ is the free abelian group generated by the isomorphy classes of vector bundles on $X$, modulo the relations of type $[E]=[E^{\prime}]-[E^{\prime \prime}]$, where $E$, $E^\prime$, and $ E^{\prime \prime}$ fit into an exact sequence $0\rightarrow E^\prime \rightarrow E \rightarrow E^{\prime \prime} \rightarrow 0$. The tensor product turns $K^{\circ}(X)$ into a ring, the Grothendieck ring of vector bundles on $X$. By Riemann-Roch, the Chern character defines a ring isomorphism $\text{ch}: K^{\circ}(X)_{\mathbb Q} \overset{\simeq}\longrightarrow A^*(X) _{\mathbb Q}$, where $A^*(X)$ is obtained by considering cycles modulo rational equivalence. Since rational equivalence is (much) finer than numerical equivalence, we obtain a ring epimorphism

\[\text{ch}: K^{\circ}(X)_{\mathbb Q} \overset{\simeq}\longrightarrow A^*(X) _{\mathbb Q}\twoheadrightarrow N^*(X)_{\mathbb Q}.\]

In many cases, we won't be able to compute the entire Chow ring: We will only be able to obtain information on a certain subring generated by some tautological classes. Therefore, we are then actually working with the class of all varieties sharing the same piece of tautological ring. We illustrate this by an example:

julia> P3 = abstract_projective_space(3)
AbstractVariety of dim 3

julia> S1 = zero_locus_section(OO(P3, 3)) # cubic hypersurface in P3
AbstractVariety of dim 2

julia> basis(S1)  # only class in codimension 1 is hyperplane class
3-element Vector{Vector{MPolyQuoRingElem}}:
 [1]
 [h]
 [h^2]

julia> P2 = abstract_projective_space(2);

julia> S2 = blow_up_points(P2, 6) # construct surface S2 by blowing up P2 in 6 points
AbstractVariety of dim 2

julia> basis(S2) # more classes in codimension 1 here
3-element Vector{Vector{MPolyQuoRingElem}}:
 [1]
 [h, e[1], e[2], e[3], e[4], e[5], e[6]]
 [h^2]

julia> e, h = gens(S2)[1:6], gens(S2)[end];

julia> H = 3h - sum(e); # embeds S2 as cubic hypersurface into P3:

julia> integral(H^2)
3

julia> euler_characteristic(OO(S2, H))
4

julia> f = map(S2, S1, [H]) # is isomorphism:
AbstractVarietyMap from AbstractVariety of dim 2 to AbstractVariety of dim 2

julia> dim(f)
0

julia> chern_character(tangent_bundle(f))
0

Although f is an isomorphism, applying the constructed pushforward map to the classes e[i] will yield a warning and not give a correct answer.

General textbooks offering details on theory and algorithms include:

• [EH16]
• [Ful98]

For the Chow rings of abstract flag bundles see

• [GSS22].

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:

• Pieter Belmans,
• Wolfram Decker,
• Tommy Hofmann.

You can ask questions in the OSCAR Slack.

Alternatively, you can raise an issue on github.