Non-μ-ordinary smooth cyclic covers of P¹

Given a family of cyclic covers of P¹ 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 ({ almost} μ-ordinary). Our method relies on further study of the extended Hasse-Witt matrix initiated in [12].