Chow ring of Hilbert scheme of 4 points in $\mathbb{P}^2$

Question

What is known about the Chow ring of the Hilbert scheme of length 4 subschemes of $\mathbb{P}^2$?

I know there is work on cycles on Hilbert schemes in the literature, but I don't know what can be deduced about this special case.

Answer 1

In principle, the Chow rings of Hilbert schemes of length $d$ subschemes in $\mathbb{P}^2$ are known (though it may still be a nontrivial task to extract information from the known descriptions). Here are some literature references. (Note that some of these talk about integral cohomology or homology, but because of the Bialynicki-Birula cell structure the cycle class map from Chow ring to integral cohomology ring is an isomorphism.)

There is one description of the Chow rings of Hilbert schemes in terms of the representation theory of the infinite-dimensional Heisenberg algebra. This is due to Nakajima and Grojnowski (after work of many people, check out the references in the papers)

• H. Nakajima: Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. of Math. 145 (1997), 379-388. link to arXiv paper

• I. Grojnowski: Instantons and affine algebras I. The Hilbert scheme and vertex operators. Math. Res. Lett. 3 (1996), 275-291. link to arXiv paper

There is another description based on computations with equivariant cohomology in the following paper (which also contains an explicit computation for the Hilbert scheme of 3 points on $\mathbb{P}^2$):

• L. Evain: The Chow ring of punctual Hilbert schemes on toric surfaces. Transformation Groups 12 (2007), 227--249. link to arXiv paper

A different basis for the integral cohomology (given by explicit geometric configurations) was given in the following paper

• R. Mallavibarrena and I. Sols. Bases for the homology groups of the Hilbert scheme of points in the plane. Compositio Math. 74 (1990), 169-201. link to numdam

Section 5 of this paper also contains some computations of intersection products in the Chow ring of the Hilbert scheme of 4 points. This may be the most relevant for the question, showing that computations of intersection products can be made explicit.

There are also some lecture notes on these results:

• H. Nakajima: Lectures on Hilbert schemes of points on surfaces. University Lecture Series 18. Amer. Math. Soc. 1999.

• G. Ellingsrud and L. Göttsche: Hilbert schemes of points and Heisenberg algebras. link to ICTP website

---

Edit: Some more references. There is of course

• L. Göttsche. Hilbert schemes of zero-dimensional subschemes of smooth varieies. Lectures Notes in Math. 1572. Springer 1994.

In the references of Göttsche's book I found the following which also provides a computation of the intersection pairing on ${\rm CH}^4$, the middle dimension of the Chow ring for ${\rm Hilb}^4(\mathbb{P}^2)$ (results are obtained by using specific geometric features, not by specialization from results for general ${\rm Hilb}^d$).

• D. Avritzer and I. Vainsencher. ${\rm Hilb}^4(\mathbb{P}^2)$. In: Enumerative Geometry (Sitges 1987), Lecture Notes in Math. 1436. Springer 1990, pp. 30-59.