On the finiteness of loci of weighted plane curves in the moduli space

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 $Z_Q\subseteq M_g$.