Deforming cyclic covers in towers

Abstract

In this paper, we study the deformations of cyclic covers over a complete discrete valuation ring of equal characteristic p > 0. We first introduce the towers of Hurwitz trees notion, which arises from given towers of cyclic covers of a rigid disc. We then apply the Hurwitz tree technique to prove that one can always extend an equal characteristic deformation of a cyclic cover's subcover to a deformation of the whole cover. That shows the exact analog of the refined lifting conjecture for cyclic covers (conjectured by Saïdi) in equal characteristic is true.