The genus of curves on the three dimensional quadric

By means of an 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.