Explicit Solution By Radicals, Gonal Maps and Plane Models of Algebraic Curves of Genus 5 or 6

TL;DR

This paper develops explicit algorithms to construct minimal degree ramified covers and plane models for algebraic curves of genus 5 and 6, extending prior work and exploring geometric properties related to gonality.

Contribution

It provides new computational methods for genus 5 and 6 curves, completing the radical parametrization framework for these cases and analyzing their plane models and gonality.

Findings

Constructed minimal degree ramified covers of degree ≤ 4 for genus 5 and 6 curves.

Established relationships between degree 6 plane models and gonality of genus 6 curves.

Extended radical parametrization techniques to higher genus curves.

Abstract

We give explicit computational algorithms to construct minimal degree (always $\le 4$) ramified covers of $\Prj^1$ for algebraic curves of genus 5 and 6. This completes the work of Schicho and Sevilla (who dealt with the $g \le 4$ case) on constructing radical parametrisations of arbitrary genus $g$ curves. Zariski showed that this is impossible for the general curve of genus $\ge 7$. We also construct minimal degree birational plane models and show how the existence of degree 6 plane models for genus 6 curves is related to the gonality and geometric type of a certain auxiliary surface.