2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/157684We study the existence of linear series on curves lying on an Enriques surface and general in their complete linear system. Using a method that works also below the Bogomolov-Reider range, we compute, in all cases, the gonality of such curves. We also give a new result about the positive cone of line bundles on an Enriques surface and we show how this relates to the gonality.To appear on Math.ZAlgebraic GeometryCommutative Algebra14H51, 14C20, 14J28text