Computational aspects of gonal maps and radical parametrization of curves

TL;DR

This paper presents algorithms for computing gonal maps of projective curves, characterizes certain curves via Betti numbers, and provides an efficient method for radical parametrization of low-gonality curves.

Contribution

It introduces a new algorithm for finding minimal degree gonal maps using scrollar syzygies and extends to radical parametrization for curves with gonality up to 4.

Findings

Algorithm successfully computes gonal maps for given curves.

Characterization of curves with unique gonal maps via Betti numbers.

Efficient radical parametrization method for curves with gonality ≤ 4.

Abstract

We develop in this article an algorithm that, given a projective curve $C$, computes a \textit{gonal map}, that is, a finite morphism from $C$ to the projective line of minimal degree. Our method is based on the computation of scrollar syzygies of canonical curves. We develop an improved version of our algorithm for curves with a unique gonal map and we discuss a characterization of such curves in terms of Betti numbers. Finally, we derive an efficient algorithm for radical parametrization of curves of gonality $\le 4$.