In mathematics, the gonality of an algebraic curve C is defined as the lowest degree of a nonconstant rational map from C to the projective line.
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.
The gonality conjecture, of M. Green and R. Lazarsfeld, predicts that the gonality of the algebraic curve C can be calculated by homological algebra means, from a minimal resolution of an invertible sheaf of high degree.
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 According to the 1900 ICM talk of Federico Amodeo, the notion (but not the terminology) originated in Section V of Riemann's Theory of Abelian Functions.