A canonical linear system associated to adjoint divisors in characteristic $p>0$

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 $\mathrm{\mid }{K}_X+L\mathrm{\mid }$. We further generalize this to incorporate a boundary divisor $\mathrm{\Delta }$. We show that these subsystems behave like the global sections associated to multiplier ideals, $H^0(X,\text{mJ}(X,\mathrm{\Delta })\text{tensor}L)$ 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.