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 Sn on n letters permuting the factors.
∪ {∞} ≈ S2), its nth symmetric product ΣnC can be identified with complex 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 divisors.
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[1] 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).