Heuristics for (ir)reducibility of $p$-rank strata of the moduli space of hyperelliptic curves
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916782734835712 |
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| author | Bouchet, Thomas Davis, Erik Groen, Steven R. Porat, Zachary York, Benjamin |
| author_facet | Bouchet, Thomas Davis, Erik Groen, Steven R. Porat, Zachary York, Benjamin |
| contents | 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 $\mathcal{H}^0_g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06457 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Heuristics for (ir)reducibility of $p$-rank strata of the moduli space of hyperelliptic curves Bouchet, Thomas Davis, Erik Groen, Steven R. Porat, Zachary York, Benjamin Algebraic Geometry Number Theory 14Q05, 14H10, 14G15, 14G17 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 $\mathcal{H}^0_g$. |
| title | Heuristics for (ir)reducibility of $p$-rank strata of the moduli space of hyperelliptic curves |
| topic | Algebraic Geometry Number Theory 14Q05, 14H10, 14G15, 14G17 |
| url | https://arxiv.org/abs/2506.06457 |