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Gonality of an algebraic curve

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Inmathematics, thegonality of analgebraic curveC is defined as the lowest degree of a nonconstantrational map fromC to theprojective line. In more algebraic terms, ifC is defined over thefieldK andK(C) denotes thefunction field ofC, then the gonality is the minimum value taken by the degrees offield extensions

K(C)/K(f)

of the function field over itssubfields generated by single functionsf.

IfK is algebraically closed, then the gonality is 1 precisely for curves ofgenus 0. The gonality is 2 for curves of genus 1 (elliptic curves) and forhyperelliptic curves (this includes all curves of genus 2). For genusg ≥ 3 it is no longer the case that the genus determines the gonality. The gonality of a curve of genusg is less than or equal to thefloor 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 thePicard curves, of genus three and given by an equation

y³ =Q(x)

whereQ is of degree 4.

Thegonality conjecture, of M. Green and R. Lazarsfeld, predicts that the gonality of the algebraic curveC can be calculated byhomological algebra means, from aminimal resolution of aninvertible sheaf of high degree. In many cases the gonality is two more than theClifford index. TheGreen–Lazarsfeld conjecture is an exact formula in terms of thegraded Betti numbers for a degreed embedding inr dimensions, ford large with respect to the genus. Writingb(C), with respect to a given such embedding ofC and the minimal free resolution for itshomogeneous coordinate ring, for the minimum indexi 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 ofFederico Amodeo, the notion (but not the terminology) originated in Section V ofRiemann'sTheory of Abelian Functions. Amodeo used the term "gonalità" as early as 1893.

References

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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.

Geometric introduction to trigonal curves of genus five

Code for constructing examples of special trigonal curves on GitHub, written inMacaulay2

Topics inalgebraic curves

Rational curves

Elliptic curves

Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

Plane curves

Riemann surfaces

Constructions

Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem (Algebraic curves)
Singularities	A_k singularity Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

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