In mathematics, the gonality of an algebraic curve C is defined as the lowest degree of a rational map from C to the projective line, which is not constant. In more algebraic terms, if C is defined over the field K and K(C) denotes the function field of C, then the gonality is the minimum value taken by the degrees of field extensions
of the function field over its subfields generated by single functions f.
The gonality is 1 precisely for curves of genus 0. It is 2 just for the hyperelliptic curves, including elliptic curves. For genus g ≥ 3 it is no longer the case that the genus determines the gonality. The gonality of the generic curve of genus g is the floor function of
Trigonal curves are those with gonality 3, and this case gave rise to the name in general. Trigonal curves include the Picard curves, of genus three and given by an equation
where Q is of degree 4.
The gonality conjecture, of M. Green and R. Lazarsfeld, predicts that the gonality of C can be calculated by homological algebra means, from a minimal resolution of an invertible sheaf of high degree. In many cases the gonality is two less than the Clifford index. The Green-Lazarsfeld conjecture is an exact formula in terms of the graded Betti numbers for a degree d embedding in r dimensions, for d large with respect to the genus. Writing b(C), with respect to a given such embedding of C and the minimal free resolution for its homogeneous coordinate ring, for the minimum index i for which βi, i + 1 is zero, then the conjectured formula for the gonality is