2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/150189By means of an {\it ad hoc} modification of the so-called ``Castelnuovo-Harris analysis" we derive an upper bound for the genus of integral curves on the three dimensional nonsingular quadric which lie on an integral surface of degree $2k$, as a function of $k$ and the degree $d$ of the curve. In order to obtain this we revisit the Uniform Position Principle to make its use computation-free. The curves which achieve this bound can be conveniently characterized. Keywords: Castelnuovo-Harris Bound, Uniform Position Principle, Low Codimension, LinkageAmsppt,15 pages, to appear in Nagoya Math. JAlgebraic Geometry14E35, 14H45, 14M06,14M07, 14M17, 14N99text