The genus of curves on the three dimensional quadric

By 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, Linkage