mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the gonality of an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

''C'' is defined as the lowest degree of a nonconstant

rational map In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal ...

from ''C'' to the

projective line In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...

. In more algebraic terms, if ''C'' is defined over the

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''K'' and ''K''(''C'') denotes the function field of ''C'', then the gonality is the minimum value taken by the degrees of

field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...

s :''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 Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

0. The gonality is 2 for curves of genus 1 (

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

s) and for

hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...

s (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 In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...

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 Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

means, from a minimal resolution of an

invertible sheaf In mathematics, an invertible sheaf is a sheaf on a ringed space that has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their intera ...

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 In algebraic geometry, the homogeneous coordinate ring is a certain commutative ring assigned to any projective variety. If ''V'' is an algebraic variety given as a subvariety of projective space of a given dimension ''N'', its homogeneous coordina ...

, 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 Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...

's ''Theory of Abelian Functions.'' Amodeo used the term "gonalità" as early as 1893.

References

*
Geometric introduction to trigonal curves of genus five

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

Macaulay2 Macaulay2 is a free computer algebra system created by Daniel Grayson (from the University of Illinois at Urbana–Champaign) and Michael Stillman (from Cornell University) for computation in commutative algebra and algebraic geometry. Overview ...

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