Wild ramification of nilpotent coverings and coverings of bounded degree

TL;DR

This paper establishes a criterion for tameness of finite étale covers of algebraic varieties based on the structure of their Galois groups, linking tameness to ramification behavior with respect to specific compactifications.

Contribution

It proves that covers with Galois groups containing a bounded index nilpotent subgroup are tame if and only if tameness holds at a single, specially chosen partial compactification.

Findings

Characterization of tame covers via a single compactification

Connection between Galois group structure and ramification behavior

Application of Temkin's uniformization and Drinfeld's Lefschetz theorem

Abstract

A finite \'etale map between irreducible, normal varieties is called tame, if it is tamely ramified with respect to all partial compactifications whose boundary is the support of a strict normal crossings divisor. We prove that if the Galois group of a Galois covering contains a normal nilpotent subgroup of index bounded by a constant N, then the covering is tame if and only if it is tamely ramified with respect to a single distinguished partial compactification only depending on N. The main tools used in the proof are Temkin's local purely inseparable uniformization and a Lefschetz type theorem due to Drinfeld.