2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65696In this paper we show that the relative canonical sheaf of a relatively minimal fibration of curves over a curve is semi-ample ; in fact, its m-tensored product is base point free for any m > 1. We use Koszul cohomology with it to prove that the relative canonical ring of the fibration is generated in degree up to five.16 pagesAlgebraic Geometry14D99text