On the generic curve of genus 3

TL;DR

This paper investigates genus 3 curves via coverings, deriving an explicit equation for the generic ternary quartic and analyzing special loci with automorphisms using computational group theory and covariants.

Contribution

It provides an explicit equation for the generic ternary quartic of genus 3 curves and characterizes degenerate loci with automorphisms using covariants.

Findings

Derived an explicit equation for the generic ternary quartic.

Identified degenerate loci corresponding to automorphism groups.

Provided covariant conditions for genus 3 curves with specific automorphisms.

Abstract

We study genus $g$ coverings of full moduli dimension of degree $d=[\frac {g+3} 2]$. There is a homomorphism between the corresponding Hurwitz space $\H$ of such covers to the moduli space $\M_g$ of genus $g$ curves. In the case $g=3$, using the signature of such covering we provide an equation for the generic ternary quartic. Further, we discuss the degenerate subloci of the corresponding Hurwitz space of such covers from the computational group theory viewpoint. In the last section, we show that one of these degenerate loci corresponds to the locus of curves with automorphism group $C_3$. We give necessary conditions in terms of covariants of ternary quartics for a genus 3 curve to belong to this locus.