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.
SERNESI E (2015). General curves on algebraic surfaces. JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK, 700, 209-233 [10.1515/crelle-2013-0019].