Deciding trigonality of algebraic curves

TL;DR

This paper introduces a method to determine if a non-hyperelliptic algebraic curve is trigonal, enabling the computation of a corresponding map, which is crucial for understanding the curve's geometric properties.

Contribution

It presents a novel algorithm based on Lie algebra techniques to decide trigonal curves and compute associated maps, advancing the analysis of algebraic curves.

Findings

The algorithm effectively identifies trigonal curves.

It computes explicit maps to the projective line.

The method integrates with radical parametrization determination.

Abstract

Let C be a non-hyperelliptic algebraic curve of genus at least 3. Enriques and Babbage proved that its canonical image is the intersection of the quadrics that contain it, except when C is trigonal (that is, it has a linear system of degree 3 and dimension 1) or C is isomorphic to a plane quintic (genus 6). We present a method to decide whether a given algebraic curve is trigonal, and in the affirmative case to compute a map from C to the projective line whose fibers cut out the linear system. It is based on the Lie algebra method presented in Schicho (2006). Our algorithm is part of a larger effort to determine whether a given algebraic curve admits a radical parametrization.