On Generators and Relations of the Rational Cohomology of Hilbert Schemes

TL;DR

This paper studies the algebraic structure of the rational cohomology of Hilbert schemes of points on the complex plane, identifying minimal generators and relations, and exploring their degrees and conjectures for general cases.

Contribution

It determines minimal generators and relations for the cohomology algebra of Hilbert schemes, providing new insights into their algebraic structure and related moduli spaces.

Findings

Identified two minimal sets of generators for the algebra

Determined degrees at which relations first occur

Proposed conjectures for the structure of relations for all d

Abstract

We consider for $d\geq 1$ the graded commutative $\mathbb{Q}$-algebra $\mathcal{A}(d):=H^*(\operatorname{Hilb}^d(\mathbb{C}^2);\mathbb{Q})$, which is also connected to the study of generalised Hurwitz spaces by work of the first author. These Hurwitz spaces are in turn related to the moduli spaces of Riemann surfaces with boundary. We determine two distinct, minimal sets of $\lfloor d/2\rfloor$ multiplicative generators of $\mathcal{A}(d)$. Additionally, we prove when the lowest degree generating relations occur. For small values of $d$ we also determine a minimal set of generating relations, which leads to several conjectures about the necessary generating relations for $\mathcal{A}(d)$.