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Suppose that $X$ is a projective variety over an algebraically closed fieldof characteristic $p > 0$. Further suppose that $L$ is an ample (or moregenerally 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 sectionsassociated to multiplier ideals, $H^0(X, \mJ(X, \Delta) \tensor L)$ incharacteristic zero. In particular, we show that these systems are in manycases base-point-free. While the original proof utilized Kawamata-Viehwegvanishing and variants of multiplier ideals, our proof uses test ideals.

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