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arXiv · 2201.13353

On Generators and Relations of the Rational Cohomology of Hilbert Schemes

Andrea Bianchi·Alexander Mangulad Christgau·Jonathan Sejr Pedersen
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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 $\mathcal{A}(d)$.

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Andrea Bianchi, Alexander Mangulad Christgau, Jonathan Sejr Pedersen. 2022-01-31. On Generators and Relations of the Rational Cohomology of Hilbert Schemes. https://doi.org/10.1007/s10801-022-01201-7

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