  Suppose that $X$ is a projective variety over an algebraically closed field
of characteristic $p > 0$. Further suppose that $L$ is an ample (or more
generally in some sense positive) divisor. We study a natural linear system in
$|K_X + L|$. We further generalize this to incorporate a boundary divisor
$\Delta$. We show that these subsystems behave like the global sections
associated to multiplier ideals, $H^0(X, \mJ(X, \Delta) \tensor L)$ in
characteristic zero. In particular, we show that these systems are in many
cases base-point-free. While the original proof utilized Kawamata-Viehweg
vanishing and variants of multiplier ideals, our proof uses test ideals.
