  In a previous article, we obtained data on the distribution of traces of
Frobenius of non-hyperelliptic genus $3$ curves over small finite fields. In
the present one, we give a heuristic explanation of these data, by
extrapolating from results on the distribution of traces of Frobenius for plane
curves whose degree is small with respect to the cardinality of their finite
base field. In particular, our methods shed some new light on the asymmetry of
the distribution around its mean value, which is related to the Serre
obstruction.
