Heuristics for (ir)reducibility of $p$-rank strata of the moduli space of hyperelliptic curves

TL;DR

This paper uses computational methods to estimate the irreducibility of p-rank strata in the moduli space of hyperelliptic curves, proposing conjectures on their geometric properties based on sampled data.

Contribution

It introduces a computational approach to estimate irreducibility of p-rank strata, leading to new conjectures on their geometric structure.

Findings

Non-ordinary locus likely geometrically irreducible for all g > 1

Moduli space _g is conjectured to be irreducible for all 1   g

Sample data supports irreducibility of _g for all 1   g

Abstract

Let $\mathcal{H}_g$ denote the moduli space of smooth hyperelliptic curves of genus $g$ in characteristic $p\geq 3$, and let $\mathcal{H}_g^f$ denote the $p$-rank $f$ stratum of $\mathcal{H}_g$ for $0 \leq f \leq g$. Achter and Pries note in their 2011 work that determining the number of irreducible components of $\mathcal{H}_g^f$ 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 $\mathcal{H}^{g-2}_g$ is irreducible and suggests that $\mathcal{H}^f_g$ is irreducible for all $1 \leq f \leq g$. We conclude with a brief discussion on $\mathcal{H}^0_g$.