Currently available as a preprint and as an official version.

The gonality of a smooth 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 ≥ 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 ≥ 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.