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

Abstract

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, $H^0(X, \mJ(X, Δ) \tensor L)$ 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.

15 pages, improved exposition and typos corrected, to appear in Journal für die reine und angewandte Mathematik