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

Abstract

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 | $\vert K_X + L\vert $ . We further generalize this to incorporate a boundary divisor Δ. We show that these subsystems behave like the global sections associated to multiplier ideals, H 0 ( X , 𝒥 ( X , Δ ) ⊗ L ) $H^0(X, \mathcal {J}(X, \Delta ) \otimes L)$ 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.