On base point freeness in positive characteristic

Abstract

We prove that if (X, A + B) is a pair defined over an algebraically closed field of positive characteristic such that (X, B) is strongly F -regular, A is ample and K X + A + B is strictly nef, then K X + A + B is ample. Similarly, we prove that for a log pair (X, A + B) with A being ample and B effective, K X + A + B is big if it is nef and of maximal nef dimension. As an application, we establish a rationality theorem for the nef threshold and various results towards the minimal model program in dimension three in positive characteristic.

Iitaka’s $C_{n,m}$ conjecture for $3$-folds in positive characteristic

“…Existence of (log) minimal models of 3-folds in positive characteristic p > 5 was first proved for canonical singularities by Hacon and Xu [24], and in general by Birkar [7] (see [44] for the lc case). The result on Mori fiber spaces was proved for terminal singularities by Cascini, Tanaka and Xu [13], and in general by Birkar and Waldron [8]. We collect some results in the following theorem, which will be used in our proof.…”

Abundance for non‐uniruled 3‐folds with non‐trivial Albanese maps in positive characteristics

“…A natural problem is abundance: for a minimal klt pair , is semi‐ample? The answer is positive when is big or is big [, , ]. This paper proves abundance for a non‐uniruled 3‐folds with non‐trivial Albanese maps.…”

Effective bounds on singular surfaces in positive characteristic

“…As far as we know, after the paper of Cascini, Tanaka and Xu had been announced, no one has yet applied their technique of constructing F -pure centers. We believe that down-to-earth examples provided in our paper may be suitable as a gentle introduction to some parts of their prolific paper, [4].…”