Hilbert Series for Configuration Spaces of Punctured Surfaces

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