mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 c ...

''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 de ...

from ''C'' to the

projective line In mathematics, 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 resultant elimination of special cases; ...

. 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 E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' 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 ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

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 ...

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'' dist ...

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 and computer science, 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 int ...

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 curve Picard may refer to: *Picardy, a region of France *Picard language, a language of France *Jean-Luc Picard, a fictional character in the ''Star Trek'' franchise Places * Picard, California, USA * Picard, Quebec, Canada * Picard (crater), a lunar ...

s, 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 coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...

of high degree. In many cases the gonality is two more than the

Clifford index In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve ''C''. Statement A divisor on a Riemann surface ''C'' is a formal sum \textstyle D = \sum_P ...

. The Green–Lazarsfeld conjecture is an exact formula in terms of the

graded Betti number In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X'N'' ...

s 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 ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X'N'' ...

, 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 Federico Amodeo (8 October 1859, Avellino – 3 November 1946, Naples) was an Italian mathematician, specializing in projective geometry, and a historian of mathematics. He received in 1883 his Ph.D. (''laurea'') in mathematics from the University ...

, the notion (but not the terminology) originated in Section V of

Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...

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

References

* {{Algebraic curves navbox Algebraic curves Homological algebra