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

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 ) ramified covers of for algebraic curves of genus 5 and 6. This completes the work of Schicho and Sevilla (who dealt with the case) on constructing radical parametrisations of arbitrary genus curves. Zariski showed that this is impossible for the general curve of genus . 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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Polynomial and algebraic computation · Commutative Algebra and Its Applications
