Abundance for threefolds in positive characteristic when $\nu=2$

TL;DR

This paper proves the abundance conjecture for certain threefolds in positive characteristic, showing that nef divisors with numerical dimension 2 are semiample.

Contribution

It establishes the semi-ampleness of nef divisors with numerical dimension 2 on threefolds over fields of characteristic greater than 3, confirming a case of the abundance conjecture.

Findings

Proves semi-ampleness of $K_X+B$ when $ u(K_X+B)=2$ for threefold pairs over characteristic p>3.

Confirms the abundance conjecture in this specific case for positive characteristic.

Advances understanding of the minimal model program in positive characteristic.

Abstract

In this paper, we prove the abundance conjecture for threefolds over a perfect field $k$ of characteristic $p > 3$ in the case of numerical dimension equals to $2$. More precisely, we prove that if $(X,B)$ be a projective lc threefold pair over $k$ such that $K_{X}+B$ is nef and $\nu(K_{X}+B)=2$, then $K_{X}+B$ is semiample.