In
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'' is defined as the lowest degree of a nonconstant
rational map 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 ''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 living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
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
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology ...
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
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 r ...
's ''Theory of Abelian Functions.'' Amodeo used the term "gonalità" as early as 1893.
References
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