Effective radical parametrization of trigonal curves

TL;DR

This paper introduces a method to determine if a non-hyperelliptic algebraic curve is trigonal and, if so, to explicitly compute the associated map to the projective line, aiding in the curve's parametrization.

Contribution

It provides a novel approach for identifying trigonal curves and computing their canonical maps, addressing a gap in the explicit parametrization of such curves.

Findings

Method successfully distinguishes trigonal curves from others.

Algorithm computes explicit maps for trigonal curves.

Enhances understanding of curve parametrization in algebraic geometry.

Abstract

Let $C$ be a non-hyperelliptic algebraic curve. It is known 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 isomorphic to a plane quintic (genus 6). In this context, 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.