2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68628Let $(X, Δ)$ be a four-dimensional log variety that is projective over the field of complex numbers. Assume that $(X, Δ)$ is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of "log quasi-numerically positive", by relaxing that of "numerically positive". Next we prove that, if the log canonical divisor $K_X + Δ$ is log quasi-numerically positive on $(X, Δ)$ then it is semi-ample.4 pages, LaTeX2e, the published versionAlgebraic Geometry14E30text