Gonality of an algebraic curve
Encyclopedia
In mathematics
, the gonality of an algebraic curve
C is defined as the lowest degree of a rational map from C to the projective line
, which is not constant. 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
s
of the function field over its subfields generated by single functions f.
The gonality is 1 precisely for curves of genus
0. It is 2 just for the hyperelliptic curves, including elliptic curve
s. 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
/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
where Q is of degree 4.
The gonality conjecture, of M. Green and R. Lazarsfeld, predicts that the gonality of 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 less 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 βi, i + 1 is zero, then the conjectured formula for the gonality is
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the gonality of an algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
C is defined as the lowest degree of a rational map from C to the projective line
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
, which is not constant. In more algebraic terms, if C is defined over the field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K and K(C) denotes the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
of C, then the gonality is the minimum value taken by the degrees of field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
s
- K(C)/K(f)
of the function field over its subfields generated by single functions f.
The gonality is 1 precisely for curves of genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
0. It is 2 just for the hyperelliptic curves, including elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s. 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
Floor function
In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...
of
/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
- y3 = Q(x)
where Q is of degree 4.
The gonality conjecture, of M. Green and R. Lazarsfeld, predicts that the gonality of C can be calculated by homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
means, from a minimal resolution of an invertible sheaf
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 OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...
of high degree. In many cases the gonality is two less 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
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...
, for the minimum index i for which βi, i + 1 is zero, then the conjectured formula for the gonality is
- r + 1 − b(C).