Bounding ramification by covers and curves

TL;DR

The paper establishes bounds on ramification of local systems on smooth varieties over fields of positive characteristic, showing they can be tamed outside codimension two by finite covers and that certain monodromy properties are preserved by curves.

Contribution

It introduces bounds on ramification and demonstrates the existence of curves that preserve monodromy and satisfy Lefschetz properties for local systems.

Findings

Local systems of bounded rank and ramification are tamified outside codimension 2 by finite covers.

In rank one, a curve preserves the monodromy of local systems.

Existence of a curve over an algebraic closure of a transcendental extension satisfying Lefschetz theorem.

Abstract

We prove that $\bar {\mathbb Q}_\ell$-local systems of bounded rank and ramification on a smooth variety $X$ defined over an algebraically closed field $k$ of characteristic $p\neq \ell$ are tamified outside of codimension $2$ by a finite separable cover of bounded degree. In rank one, there is a curve which preserves their monodromy. There is a curve defined over the algebraic closure of a purely transcendental extension of $k$ of finite degree which fulfills the Lefschetz theorem. Last version: minor typos corrected.