  We give upper bounds on the genus of a curve with general moduli assuming
that it can be embedded in a projective nonsingular surface $Y$ so that
$\dim(|C|) > 0$. We find such bounds for all types of surfaces of intermediate
Kodaira dimension and, under mild restrictions, for surfaces of general type
whose minimal model $Z$ satisfies the Castelnuovo inequality $K_Z^2 \ge
3\chi(\O_Z) - 10$. In this last case we obtain $g \le 19$. In the other cases
considered the bounds are lower.
