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

TL;DR

This paper investigates the loci of smooth weighted plane curves of fixed genus in the moduli space, showing infinitely many weight quadruples yield such curves, but only finitely many distinct loci in the moduli space.

Contribution

It proves the finiteness of loci of weighted plane curves of fixed genus in the moduli space despite infinitely many weight quadruples.

Findings

Infinitely many quadruples produce curves of fixed genus

Finitely many loci correspond to these quadruples in the moduli space

Loci are distinct for different quadruples despite infinite quadruples

Abstract

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 $Z_Q\subseteq M_g$.