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

Abstract

For every fixed genus $g\ge 1$, we consider all quadruples $Q=(w_0,w_1,w_2,d)\in {\mathbb{Z}}_{>0}^4$ 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 $Z_Q\subseteq M_g$.

17 pages