Non-$\mu$-ordinary smooth cyclic covers of ${\mathbb{P}}^1$

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].