Gonality of an algebraic curve
- 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, 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
:"K"("C")/"K"("f")
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
:("g" + 3)/2.
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 "C" can be calculated by homological algebra means, from a minimal resolution of an invertible sheaf of high degree. See Koszul cohomology.
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