Papers › Heuristics for (ir)reducibility of p-rank strata of the moduli space of hyperelliptic curves
Heuristics for (ir)reducibility of p-rank strata of the moduli space of hyperelliptic curves
Thomas Bouchet, Erik Davis, Steven R. Groen, Zachary Porat, Benjamin York
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Let ℋ_g denote the moduli space of smooth hyperelliptic curves of genus g in characteristic p≥3, and let ℋ_gᶠ denote the p-rank f stratum of ℋ_g for 0 ≤f ≤g. Achter and Pries note in their 2011 work that determining the number of irreducible components of ℋ_gᶠ 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 ℋᵍ⁻²_g is irreducible and suggests that ℋᶠ_g is irreducible for all 1 ≤f ≤g. We conclude with a brief discussion on ℋ⁰_g.
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