  For every fixed genus $g\geq 1$, we consider all quadruples
$Q=(w_0,w_1,w_2,d)\in\mathbb{Z}^4_{>0}$ with the property that any smooth
degree-$d$ curve embedded in the weighted projective plane
$\mathbb{P}^2(w_0,w_1,w_2)$ has genus $g$. We show there are infinitely many
quadruples $Q$ satisfying this condition. For every such $Q$, we consider
$Z_Q\subseteq M_g$ the locus in the moduli space of all smooth degree-$d$
curves embedded in $\mathbb{P}^2(w_0,w_1,w_2)$. We show that, as $Q$ varies
over all these quadruples, there are only finitely many different loci
$Z_Q\subseteq M_g$.
