Papers › On the Maximum Gonality of a Curve over a Finite Field
On the Maximum Gonality of a Curve over a Finite Field
Xander Faber, Jon Grantham, Everett W. Howe
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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 ≥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.
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