Abundance for slc surfaces over arbitrary fields

TL;DR

This paper proves the abundance conjecture for slc surfaces over any field of positive characteristic, extending previous results and applying advanced techniques to broader classes of surfaces and threefolds.

Contribution

It establishes the abundance conjecture for projective slc surfaces over arbitrary fields of positive characteristic, using descent techniques and previous results on lc surfaces.

Findings

Proves abundance conjecture for slc surfaces in positive characteristic.

Extends results to dlt threefold pairs and mixed characteristic families.

Utilizes descent of semi-ampleness from normalization.

Abstract

We prove the abundance conjecture for projective slc surfaces over arbitrary fields of positive characteristic. The proof relies on abundance for lc surfaces over abritrary fields, proved by Tanaka, and on the technique of Hacon and Xu to descend semi-ampleness from the normalization. We also present applications to dlt threefold pairs, and to mixed characteristic families of surfaces.