Symmetric product of an algebraic curve
Encyclopedia
In mathematics
, the n-fold symmetric product of an algebraic curve
C is the quotient space
of the n-fold cartesian product
or Cn by the group action
of the symmetric group
on n letters permuting the factors. It exists as a smooth algebraic variety
ΣnC; if C is a compact Riemann surface
it is therefore a complex manifold
. Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on C of degree n, that is, formal sums of points with non-negative integer coefficients.
For C the projective line
(say the Riemann sphere
) ΣnC can be identified with projective space
of dimension n.
If G has genus
g ≥ 1 then the ΣnC are closely related to the Jacobian variety
J of C. More accurately for n taking values up to g they form a sequence of approximations to J from below: their images in J under addition on J (see theta-divisor
) have dimension n and fill up J, with some identifications caused by special divisor
s.
For g = n we have ΣgC actually birationally equivalent to J; the Jacobian is a blowing down
of the symmetric product. That means that at the level of function field
s it is possible to construct J by taking linearly disjoint
copies of the function field of C, and within their compositum taking the fixed subfield of the symmetric group. This is the source of André Weil
's technique of constructing J as an abstract variety
from 'birational data'. Other ways of constructing J, for example as a Picard variety, are preferred now (Greg W. Anderson (Advances in Math.172 (2002) 169–205) provided an elementary construction as lines of matrices). But this does mean that for any rational function F on C
makes sense as a rational function on J, for the xi staying away from the poles of F.
For N > g the mapping from ΣnC to J by addition fibers it over J; when n is large enough (around twice g) this becomes a projective space
bundle (the Picard bundle). It has been studied in detail, for example by Kempf and Mukai.
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 n-fold symmetric product 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 the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of the n-fold cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
- C × C × ... × C
or Cn by the group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
on n letters permuting the factors. It exists as a smooth algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
ΣnC; if C is a compact Riemann surface
Compact Riemann surface
In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact and the open .A compact Riemann surface C that is a...
it is therefore a complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
. Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on C of degree n, that is, formal sums of points with non-negative integer coefficients.
For C 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...
(say the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
) ΣnC can be identified with projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
of dimension n.
If G has 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...
g ≥ 1 then the ΣnC are closely related to the Jacobian variety
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
J of C. More accurately for n taking values up to g they form a sequence of approximations to J from below: their images in J under addition on J (see theta-divisor
Theta-divisor
In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers by the zero locus of the associated Riemann theta-function...
) have dimension n and fill up J, with some identifications caused by special divisor
Special divisor
In mathematics, in the theory of algebraic curves, certain divisors on a curve C are particular, in the sense of determining more compatible functions than would be predicted. These are the special divisors...
s.
For g = n we have ΣgC actually birationally equivalent to J; the Jacobian is a blowing down
Blowing down
In mathematics, blowing down is a type of geometric modification in algebraic geometry. It is the inverse operation of blowing up.On an algebraic surface, blowing down a curve lying on the surface is a typical effect of a birational transformation...
of the symmetric product. That means that at the level of function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
s it is possible to construct J by taking linearly disjoint
Linearly disjoint
In mathematics, algebras A, B over a field k inside some field extension \Omega of k are said to be linearly disjoint over k if the following equivalent conditions are met:...
copies of the function field of C, and within their compositum taking the fixed subfield of the symmetric group. This is the source of André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
's technique of constructing J as an abstract variety
Abstract variety
In mathematics, in the field of algebraic geometry, the idea of abstract variety is to define a concept of algebraic variety in an intrinsic way. This followed the trend in the definition of manifold independent of any ambient space by some years, the first notions being those of Oscar Zariski and...
from 'birational data'. Other ways of constructing J, for example as a Picard variety, are preferred now (Greg W. Anderson (Advances in Math.172 (2002) 169–205) provided an elementary construction as lines of matrices). But this does mean that for any rational function F on C
- F(x1) + ... + F(xg)
makes sense as a rational function on J, for the xi staying away from the poles of F.
For N > g the mapping from ΣnC to J by addition fibers it over J; when n is large enough (around twice g) this becomes a projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
bundle (the Picard bundle). It has been studied in detail, for example by Kempf and Mukai.