Non-$μ$-ordinary smooth cyclic covers of $\mathbb{P}^1$
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| Format: | Preprint |
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2024
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| _version_ | 1866916286279188480 |
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| author | Lin, Yuxin Mantovan, Elena Singhal, Deepesh |
| author_facet | Lin, Yuxin Mantovan, Elena Singhal, Deepesh |
| contents | Given a family of cyclic covers of $\mathbb{P}^1$ and a prime $p$ of good reduction, by [12] the generic Newton polygon (resp. Ekedahl--Oort type) in the family ($μ$-ordinary) is known. In this paper, we investigate the existence of non-$μ$-ordinary smooth curves in the family. In particular, under some auxiliary conditions, we show that when $p$ is sufficiently large the complement of the $μ$-ordinary locus is always non empty, and for $1$-dimensional families with condition on signature type, we obtain a lower bound for the number of non-$μ$-ordinary smooth curves. In specific examples, for small $m$, the above general statement can be improved, and we establish the non emptiness of all codimension 1 non-$μ$-ordinary Newton/Ekedahl--Oort strata ({\em almost} $μ$-ordinary). Our method relies on further study of the extended Hasse-Witt matrix initiated in [12]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09632 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-$μ$-ordinary smooth cyclic covers of $\mathbb{P}^1$ Lin, Yuxin Mantovan, Elena Singhal, Deepesh Algebraic Geometry Number Theory Given a family of cyclic covers of $\mathbb{P}^1$ and a prime $p$ of good reduction, by [12] the generic Newton polygon (resp. Ekedahl--Oort type) in the family ($μ$-ordinary) is known. In this paper, we investigate the existence of non-$μ$-ordinary smooth curves in the family. In particular, under some auxiliary conditions, we show that when $p$ is sufficiently large the complement of the $μ$-ordinary locus is always non empty, and for $1$-dimensional families with condition on signature type, we obtain a lower bound for the number of non-$μ$-ordinary smooth curves. In specific examples, for small $m$, the above general statement can be improved, and we establish the non emptiness of all codimension 1 non-$μ$-ordinary Newton/Ekedahl--Oort strata ({\em almost} $μ$-ordinary). Our method relies on further study of the extended Hasse-Witt matrix initiated in [12]. |
| title | Non-$μ$-ordinary smooth cyclic covers of $\mathbb{P}^1$ |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2406.09632 |