2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143925In this note we give a quick and simple proof of the existence (and uniqueness) of Zariski decompositions on surfaces. While Zariski's original proof employs a rather sophisticated procedure to construct the negative part of the decomposition, the present approach is based on the idea that the positive part can be constructed from a maximality condition. It may also be useful that this approach yields a practical algorithm for the computation of the positive part.Algebraic Geometry14C20; 14F05text