2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/147147Given a triple covering $X$ of genus $g$ of a general (in the sense of Brill-Noether) curve $C$ of genus $h$, we show the existence of base-point-free pencils of degree $d$ which are not composed with the triple covering for any $d\ge g-[{3h+1\over 2}]-1$ by utilizing some enumerative methods and computations. We also discuss about the sharpness of our main result and the so-called Castelnuovo-Severi bound byexhibiting some examples.AmsTeX v 2.3Algebraic Geometry14H45(Primary) 14H10, 14C20(Secondary)text