Title: Non-$μ$-ordinary smooth cyclic covers of $\mathbb{P}^1$ Show Chinese title

Title: 非$μ$-平凡光滑循环覆盖的$\mathbb{P}^1$

Abstract: 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 ($\mu$-ordinary) is known. In this paper, we investigate the existence of non-$\mu$-ordinary smooth curves in the family. In particular, under some auxiliary conditions, we show that when $p$ is sufficiently large the complement of the $\mu$-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-$\mu$-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-$\mu$-ordinary Newton/Ekedahl--Oort strata ({\em almost} $\mu$-ordinary). Our method relies on further study of the extended Hasse-Witt matrix initiated in [12]. Show Chinese abstract

Abstract: 给定一个循环覆盖族$\mathbb{P}^1$和一个具有良好约化的素数$p$，根据[12]，该族中的通用牛顿多边形（或 Ekedahl--Oort 类型）（$\mu$-普通）是已知的。在本文中，我们研究该族中非$\mu$-普通光滑曲线的存在性。 特别是，在一些辅助条件下，我们证明当$p$足够大时，$\mu$-普通点集的补集总是非空的，并且对于满足符号类型条件的$1$-维族，我们得到了非$\mu$-普通光滑曲线数量的下界。 在具体例子中，对于较小的$m$，上述一般性陈述可以得到改进，我们建立了所有余维1非$\mu$-平凡的牛顿/埃克达尔-奥特结构（{\em 几乎} $\mu$ -平凡）的非空性。我们的方法依赖于对[12]中提出的扩展哈斯-维特矩阵的进一步研究。