Hurwitz trees and deformations of Artin-Schreier covers

2020/02/10 by Dang, Huy
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2002.03719

Abstract

Let R be a complete discrete valuation ring of equal characteristic p>0. Given a ℤ/p-Galois cover of a formal disc over R, one can derive from it a semi-stable model for which the specializations of branch points are distinct and lie in the smooth locus of the special fiber. The description leads to a combinatorial object which resembles a classical Hurwitz tree in mixed characteristic, which we will give the same name. The existence of a Hurwitz tree is necessary for the existence of a ℤ/p-cover whose branching data fit into that tree. We show that the conditions imposed by a Hurwitz tree's structure are also sufficient. Using this, we improve a known result about the connectedness of the moduli space of Artin-Schreier curves of fixed genus.

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