Bounding ramification by covers and curves

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 $p\ne \ell$ 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.

13 pages