# Symmetric product of an algebraic curve

In mathematics, the *n*-fold **symmetric product** of an algebraic curve *C* is the quotient space of the *n*-fold cartesian product

*C*×*C*× ... ×*C*

or *C*^{n} by the group action of the symmetric group on *n* letters permuting the factors. It exists as a smooth algebraic variety Σ^{n}*C*; 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) Σ^{n}*C* can be identified with projective space of dimension *n*.

If *G* has genus *g* ≥ 1 then the Σ^{n}*C* 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.

For *g* = *n* we have Σ^{g}*C* actually birationally equivalent to *J*; the Jacobian is a blowing down of the symmetric product. That means that at the level of function fields 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 Mathematics* 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*(*x*_{1}) + ... +*F*(*x*_{g})

makes sense as a rational function on *J*, for the *x*_{i} staying away from the poles of *F*.

For *N* > *g* the mapping from Σ^{n}*C* 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.

## Betti numbers and the Euler characteristic of the symmetric product

Let *C* be smooth and projective of genus *g* over the complex numbers **C**. The Betti numbers *b*_{i}(Σ^{n}C) of the symmetric product are given by

and the topological Euler characteristic *e*(Σ^{n}C) is given by

Here we have set *u*=-1 and *y* = - *p* in the formula before.

## References

- Macdonald, I. G. (1962), "Symmetric products of an algebraic curve",
*Topology*,**1**(4): 319–343, doi:10.1016/0040-9383(62)90019-8, MR 0151460 - Anderson, Greg W. (2002), "Abeliants and their application to an elementary construction of Jacobians",
*Advances in Mathematics*,**172**(2): 169–205, arXiv:math/0112321, doi:10.1016/S0001-8708(02)00024-5, MR 1942403