Invariants of genus 4 curves

TL;DR

This paper classifies non-hyperelliptic genus 4 curves over algebraically closed fields of characteristic 0 by explicitly describing invariants that distinguish their isomorphism classes, based on their embedding in quadrics of rank 3 or 4.

Contribution

It provides a complete set of explicit invariants (60 for rank 3 and 65 for rank 4) for classifying these curves, computed via transvectants.

Findings

60 invariants classify rank 3 curves

65 invariants classify rank 4 curves

Invariants are efficiently computable

Abstract

The present paper gives an explicit classification of the isomorphism classes of non-hyperelliptic genus 4 curves over an algebraically closed field of characteristic 0. A non-hyperelliptic genus 4 curve lies on a quadric in $\mathbb{P^3}$ of rank 3 or 4. In the case of rank 3, we give a set of 60 invariants which classify the isomorphism classes, and in the case of rank 4, we find 65 invariants. These invariants are defined by transvectants and can be efficiently computed on a given example.