  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.
