Gonality of an algebraic curve
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 that isn't 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
- K(C)/K(f)
of the function field over its subfields generated by single functions f.
If K is algebraically closed, then the gonality is 1 precisely for curves of genus 0. It is 2 for hyperelliptic curves (this includes all curves of genus 2) and curves of genus 1 (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
- (g + 3)/2.
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
- y3 = Q(x)
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 more 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
- r + 1 − b(C).
References
- Eisenbud, David (2005). The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry. Graduate Texts in Mathematics. Vol. 229. New York, NY: Springer-Verlag. pp. 171, 178. ISBN 0-387-22215-4. MR 2103875. Zbl 1066.14001.