Hurwitz trees and deformations of Artin–Schreier covers

Let R be a complete discrete valuation ring of equal characteristic p > 0 . In this paper, we first generalize the notion of Hurwitz trees in mixed characteristic to Artin–Schreier covers ( ℤ / p -covers in characteristic p ) of a formal disc over R . It is a combinatorial-differential object with the shape of the dual graph of a cover’s semi-stable model, and is endowed with essential degeneration data, measured by Kato’s refined Swan conductors, of that cover. We then show how the existence of a deformation between two given Artin–Schreier covers is completely determined by the existence of certain Hurwitz trees. Finally, using the Hurwitz trees criteria, we improve a known result about the connectedness of the moduli space of Artin–Schreier curves of fixed genus.