Homology of Hilbert schemes of reducible locally planar curves

TL;DR

This paper extends homological results of Hilbert schemes from irreducible to reducible locally planar curves by constructing an algebra acting on their homology and analyzing a specific example.

Contribution

It introduces an algebra acting on the homology of Hilbert schemes for reducible curves and embeds it into a Weyl algebra, generalizing previous results.

Findings

Constructed an algebra A acting on the homology of Hilbert schemes of reducible curves.

Realized algebra A as a subalgebra of the Weyl algebra of affine space.

Computed the representation for the case of a nodal reducible curve.

Abstract

Let $C$ be a complex, reduced, locally planar curve. We extend the results of Rennemo arXiv:1308.4104 to reducible curves by constructing an algebra $A$ acting on $V=\bigoplus_{n\geq 0} H_*(C^{[n]}, \mathbb{Q})$, where $C^{[n]}$ is the Hilbert scheme of $n$ points on $C$. If $m$ is the number of irreducible components of $C$, we realize $A$ as a subalgebra of the Weyl algebra of $\mathbb{A}^{2m}$. We also compute the representation $V$ in the simplest reducible example of a node.