Heuristics for (Ir)Reducibility of p-rank Strata of the Moduli Space of Hyperelliptic Curves

Let Hg denote the coarse moduli space of smooth hyperelliptic curves of genus g in characteristic p≥3, and let Hgf denote the p-rank f stratum of Hg for 0≤f≤g. Achter and Pries note that determining the number of irreducible components of Hgf would lead to several intriguing corollaries. In this paper, we present a computational approach for estimating the number of irreducible components in various p-rank strata. Our strategy involves sampling curves over finite fields and calculating their p-ranks. From the data gathered, we conjecture that the non-ordinary locus is geometrically irreducible for all genera g>1. The data also leads us to conjecture that the moduli space Hgg−2 is irreducible and suggests that Hgf is irreducible for all 1≤f≤g. We conclude with a brief discussion on Hg0.