Base point free theorem of Reid-Fukuda type

TL;DR

This paper proves a base point free theorem for proper dlt pairs, establishing conditions under which multiples of a nef Cartier divisor are base point free, extending the Reid-Fukuda type results.

Contribution

It introduces a new base point free theorem for dlt pairs with nef and log big divisors, generalizing previous results in the minimal model program.

Findings

For proper dlt pairs, sufficiently large multiples of certain nef divisors are base point free.

The theorem applies when a divisor is nef, Cartier, and satisfies nef and log big conditions.

Provides a criterion for base point freeness in the context of the minimal model program.

Abstract

Let $(X,\Delta)$ be a proper dlt pair and $L$ a nef Cartier divisor such that $aL-(K_X+\Delta)$ is nef and log big on $(X,\Delta)$ for some $a\in {\mathbb Z}_{>0}$. Then $|mL|$ is base point free for every $m\gg 0$.