Hilbert Series for Configuration Spaces of Punctured Surfaces

TL;DR

This paper computes the generating functions for mixed Hodge numbers of configuration spaces on punctured surfaces, revealing new stable patterns and extending previous results to all genera and punctures.

Contribution

It provides the first explicit formulas for the mixed Hodge numbers of configuration spaces on punctured surfaces for all genera and puncture counts, extending known cases.

Findings

Derived explicit generating functions for genus g≥2.

Connected classical and secondary stability phenomena in Hodge numbers.

Extended stability results to all punctured surfaces.

Abstract

Let $\Sigma_{g,r}$ denote the $r$-punctured closed Riemann surface of genus $g$. For every $g\geq 0$, we determine the four-variable generating function for the mixed Hodge numbers of the unordered configuration spaces of $\Sigma_{g,1}$. The cases where $g\geq 2$ are new. Combining a result of \cite{huang2020cohomology}, this determines the analogous generating function for $\Sigma_{g,r}$ for all $r\geq 1$. As an application of our formula we illustrate how classical homological stability results, as well as so-called secondary stability results of \cite{miller2019higher} can be interpolated to illustrate stable behaviors in the mixed Hodge numbers of these spaces which have been thus-far undiscovered.