On the maximum gonality of a curve over a finite field

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.