THE GENUS OF CURVES ON THE 3-DIMENSIONAL QUADRIC

By means of an ad hoc modification of the so-called ''Castelnuovo-Harr is analysis'' we derive an upper bound for the genus of integral curve s on the three dimensional nonsingular quadric which lie on an integra l surface of degree 2k, as a function of k and the degree d of the cur ve. 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.