On quasi-log schemes

TL;DR

This paper proves a fundamental basepoint-free theorem for quasi-log schemes, extending Ambro's foundational work, using Kawamata's X-method and basic slc-trivial fibrations, with added explanations for clarity.

Contribution

It establishes the basepoint-free theorem of Reid--Fukuda type for quasi-log schemes in full generality, confirming all results claimed in Ambro's original paper.

Findings

Proves the basepoint-free theorem for quasi-log schemes

Confirms the validity of results in Ambro's foundational work

Provides accessible explanations on quasi-log schemes

Abstract

The notion of quasi-log schemes was first introduced by Florin Ambro in his epoch-making paper: Quasi-log varieties. In this paper, we establish the basepoint-free theorem of Reid--Fukuda type for quasi-log schemes in full generality. Roughly speaking, it means that all the results for quasi-log schemes claimed in Ambro's paper hold true. The proof is Kawamata's X-method with the aid of the theory of basic slc-trivial fibrations. For the reader's convenience, we make many comments on the theory of quasi-log schemes in order to make it more accessible.