On the semiampleness of the positive part of CKM Zariski decompositions

TL;DR

This paper investigates the conditions under which graded rings associated with big divisors on LC pairs are finitely generated, focusing on the positivity at LC centers and the existence of Zariski decompositions.

Contribution

It establishes finite generation results for these rings when the pair is DLT or low-dimensional, assuming a Zariski decomposition exists and sufficient positivity.

Findings

Finite generation proven for DLT pairs.

Finite generation proven in low dimensions.

Results depend on positivity at LC centers.

Abstract

We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the pair is DLT or the dimension is low, given that a Zariski decomposition exists.