Homology of Hilbert schemes of reducible locally planar curves

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 $V={⨁}_{n\ge 0}{H}_{\ast }(C^{[n]},\mathbb{Q})$, 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.

15 pages, one table, no figures

References in corpus (2)

• Toric braids and $(m,n)$-parking functions
• Khovanov-Rozansky homology and higher Catalan sequences