Non--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 (-ordinary) is known. In this paper, we investigate the existence of non--ordinary smooth curves in the family. In particular, under some auxiliary conditions, we show that when p is sufficiently large the complement of the -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--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--ordinary Newton/Ekedahl--Oort strata (almost -ordinary). Our method relies on further study of the extended Hasse-Witt matrix initiated in 12.