View source for Symmetric product of an algebraic curve

In [[mathematics]], the ''n''-fold '''symmetric product''' of an [[algebraic curve]] ''C'' is the [[Quotient space (topology)|quotient space]] of the ''n''-fold [[cartesian product]]

:''C'' × ''C'' × ... × ''C''

or ''C''<sup>''n''</sup> by the [[Group action (mathematics)|group action]] of the [[symmetric group]] ''S''<sub>''n''</sub> on ''n'' letters permuting the factors. It exists as a [[smooth scheme|smooth]] [[algebraic variety]] denoted by Σ<sup>''n''</sup>''C''. If ''C'' is a [[compact Riemann surface]], Σ<sup>''n''</sup>''C'' is therefore a [[complex manifold]]. Its interest in relation to the classical geometry of curves is that its points correspond to [[effective divisor]]s on ''C'' of degree&nbsp;''n'', that is, [[formal sum]]s of points with non-negative integer coefficients.

For ''C'' the [[projective line]] (say the [[Riemann sphere]] <math>\mathbb{C}</math> ∪ {∞}  ≈  ''S''<sup>''2''</sup>), its nth symmetric product Σ<sup>''n''</sup>''C'' can be identified with [[projective space|complex projective space]] <math>\mathbb{CP}^n</math> of dimension&nbsp;''n''.

If ''G'' has [[genus (mathematics)|genus]] ''g''&nbsp;≥&nbsp;1 then the Σ<sup>''n''</sup>''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 divisor]]s.

For ''g'' = ''n'' we have Σ<sup>''g''</sup>''C'' actually [[birationally equivalent]] to ''J''; the Jacobian is a [[blowing up|blowing down]] of the symmetric product. That means that at the level of [[function field of an algebraic variety|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<ref>{{harvtxt|Anderson|2002}} provided an elementary construction as lines of matrices.</ref> but this does mean that for any rational function ''F'' on ''C''

:''F''(''x''<sub>1</sub>) + ... + ''F''(''x''<sub>''g''</sub>)

makes sense as a rational function on ''J'', for the ''x''<sub>''i''</sub> staying away from the poles of ''F''.

For ''n''&nbsp;>&nbsp;''g'' the mapping from Σ<sup>''n''</sup>''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 a smooth projective curve of genus ''g'' over the complex numbers '''C'''. The [[Betti number]]s ''b''<sub>''i''</sub>(Σ<sup>''n''</sup>C) of the symmetric products Σ<sup>''n''</sup>C for all n = 0, 1, 2, ... are given by the [[generating function]]
:<math> \sum_{n = 0}^{\infty} \sum_{i = 0}^{2n} b_i( \Sigma^n C ) y^n u^{i-n} = \frac{ (1 + y)^{2g} }{ (1 - u y) (1 - u^{-1} y) }</math>
and their Euler characteristics ''e''(Σ<sup>''n''</sup>C) are given by the generating function
:<math> \sum_{n = 0}^{\infty} e( \Sigma^n C ) p^n = (1 - p)^{2g - 2} . </math>
Here we have set ''u'' = -1 and ''y'' = -''p'' in the previous formula.

==Notes==
{{reflist}}

==References==

* {{Citation | last1=Macdonald | first1=I. G. | authorlink = Ian G. Macdonald | title=Symmetric products of an algebraic curve  | doi=10.1016/0040-9383(62)90019-8  | mr=0151460 | year=1962 | journal=[[Topology (journal)|Topology]] | volume=1 | issue=4 | pages=319–343| doi-access=free }}
* {{Citation | last1=Anderson | first1=Greg W. | authorlink = Greg W. Anderson | title=Abeliants and their application to an elementary construction of Jacobians | doi=10.1016/S0001-8708(02)00024-5 | doi-access=free | mr = 1942403 | year=2002 | journal = [[Advances in Mathematics]] | volume = 172 | issue = 2 | pages = 169–205| arxiv = math/0112321 }}

[[Category:Algebraic curves]]
[[Category:Symmetric functions]]