Huy Dang: Deforming cyclic covers in towers

Abstract

There are many interesting phenomena in the case of positive or mixed characteristics. For instance, unlike in characteristic zero, there exist covers of curves whose number of branch points are different but still lie in the same flat family. In this talk, we briefly discuss the process of showing that a flat equal-characteristic p deformation of a cyclic sub-covering always extends to that of the whole cover. That shows the p-fibers of the canonical maps between the moduli space of cyclic coverings and ones of the sub-coverings are surjective. The crucial technique is a study of local covering’s degeneration using Kato–Saito–Abbes’ refined Swan conductor, which generalizes the classical perfect residue case.