High Quality Content by WIKIPEDIA articles! In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian productor Cn by the group action of the symmetric group on n letters permuting the factors. It exists as a smooth algebraic variety ?nC; 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) ?nC can be identified with projective space of dimension n.

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