Log abundance for threefolds in characteristic $p>3$ in numerical dimension one

Principal Theorem 1.1: over an algebraically closed field of characteristic $p>3$, if $(X,B)$ is a projective log canonical threefold pair with effective $\mathbb Q$-boundary, $K_X+B$ is $\mathbb Q$-Cartier, nef and of numerical dimension one, then $K_X+B$ is semiample; neither terminality nor $\mathbb Q$-factoriality is needed. The companion proves the terminal $\mathbb Q$-factorial case ($K_X$ nef with $\nu=1$ gives $\kappa=1$ and semiampleness). Characteristics 2 and 3, numerical dimension zero and higher dimensions are not addressed.

Principal Theorem 1.1: over an algebraically closed field of characteristic $p>3$, if $(X,B)$ is a projective log canonical threefold pair with effective $\mathbb Q$-boundary, $K_X+B$ is $\mathbb Q$-Cartier, nef and of numerical dimension one, then $K_X+B$ is semiample; neither terminality nor $\mathbb Q$-factoriality is needed. The companion proves the terminal $\mathbb Q$-factorial case ($K_X$ nef with $\nu=1$ gives $\kappa=1$ and semiampleness). Characteristics 2 and 3, numerical dimension zero and higher dimensions are not addressed.