On the Maximum Gonality of a Curve over a Finite Field

TL;DR

This paper improves the upper bound on the gonality of curves over finite fields, showing it is at most equal to the genus for most cases, with a few explicit exceptions for low genus.

Contribution

The authors refine the known gonality bounds over finite fields using a combination of geometric and computational methods, identifying specific exceptions.

Findings

Gonality of a curve over finite fields is at most equal to its genus for g ≥ 5.

Explicit examples of curves with higher gonality for g=3 and g=4 are provided.

The paper identifies and reproduces equations for exceptional curves with higher gonality.

Abstract

The gonality of a smooth geometrically connected curve over a field $k$ is the smallest degree of a nonconstant $k$-morphism from the curve to the projective line. In general, the gonality of a curve of genus $g \ge 2$ is at most $2g - 2$. Over finite fields, a result of F.K. Schmidt from the 1930s can be used to prove that the gonality is at most $g+1$. Via a mixture of geometry and computation, we improve this bound: for a curve of genus $g \ge 5$ over a finite field, the gonality is at most $g$. For genus $g = 3$ and $g = 4$, the same result holds with exactly $217$ exceptions: There are two curves of genus $4$ and gonality $5$, and $215$ curves of genus $3$ and gonality $4$. The genus-$4$ examples were found in other papers, and we reproduce their equations here; in supplementary material, we provide equations for the genus-$3$ examples.