2022/07/28 by Xander Faber, Faber, Xander, Jon Grantham +3
Mathematics · Social Sciences · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Communism, Protests, Social Movements #FOS: Mathematics #Number Theory (math.NT) #Vietnamese History and Culture Studies
paper · pdf · doi:10.48550/arxiv.2207.14307
openalex publication_date 2022/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.