# Gonality of an algebraic curve

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. 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. The gonality is 2 for curves of genus 1 (elliptic curves) and for hyperelliptic curves (this includes all curves of genus 2). 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

*y*^{3}=*Q*(*x*)

where *Q* is of degree 4.

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. 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 β_{i, i + 1} is zero, then the conjectured formula for the gonality is

*r*+ 1 −*b*(*C*).

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.* Amodeo used the term "gonalità" as early as 1893.

## References

- Eisenbud, David (2005).
*The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry*. Graduate Texts in Mathematics.**229**. New York, NY: Springer-Verlag. pp. 171, 178. ISBN 0-387-22215-4. MR 2103875. Zbl 1066.14001.