Hilbert Series for Configuration Spaces of Punctured Surfaces

Abstract

Let ${\Sigma }_{g,r}$ denote the $r$-punctured closed Riemann surface of genus $g$. For every $g\ge 0$, we determine the four-variable generating function for the mixed Hodge numbers of the unordered configuration spaces of ${\Sigma }_{g,1}$. The cases where $g\ge 2$ are new. Combining a result of \cite{huang2020cohomology}, this determines the analogous generating function for ${\Sigma }_{g,r}$ for all $r\ge 1$. 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.