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

TL;DR

This paper investigates a natural linear system associated with adjoint divisors on projective varieties over fields of characteristic p > 0, demonstrating base-point-freeness properties using test ideals.

Contribution

It introduces a characteristic p > 0 analog of multiplier ideal-based linear systems and proves their base-point-freeness without relying on Kawamata-Viehweg vanishing.

Findings

Linear systems behave like multiplier ideal sections in characteristic zero

Many of these systems are base-point-free

Uses test ideals instead of vanishing theorems

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, $H^0(X, \mJ(X, \Delta) \tensor L)$ 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.