Reference-Request: Symmetric Product Schemes Is there a good reference for the theory of symmetric product schemes? (I only need a few basic things, the construction, etc.)
Googling it turned up a lot of papers which use it as if it's common knowledge, so I suspect there should be a reference somewhere, but I can't find any.
 A: I don't think there is a reference addressing exactly your question, so I'll just try my best to answer your question using more general knowledge and providing a number of references.


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*If we have a finite group $G$ acting on a finitely generated algebra $A$ over a field $k$, we know by the Artin-Tate lemma that $A^G$ is also noetherian (it is a finitely generated algebra over $k$ in case $A$ is).  You can find this e.g. in Atiyah & Macdonald, Introduction to Commutative Algebra.  The idea is that $A$ is an $A^G$-module of finite type.  It is easy to see that $K^G$ is the quotient field of $A^G$, in the case where $A$ is a domain.

*For actions of finite groups on schemes: when we have a scheme $S$ and a finite group $G$ acting on $X$, one may reduce with some generality to the affine case, solved above.  Indeed, this is the case for $X$ quasiprojective scheme over a ring $A$ (we will take $A$ to be Noetherian).  I know that this result is due to Michael Artin, but I saw it in James Milne's Etale Cohomology book.
I would say that a good reference in general is Mumford's Geometric Invariant Theory.
IMPORTANT: The condition that is required to ensure the reduction to the affine case is the following.  If any $G$-orbit admits an affine open set containing it, then we may construct $X/G$ by reduction to the affine case.  If $X$ is quasiprojective over an affine ring, this is the case for any finite set of points.


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*It is easy to see that, if $X$ is normal, so is $X/G$ provided $X$ is quasiprojective over a field or so (i.e. provided we fall under the above hypotheses).

*In the case of a smooth curve $C$, you may consider the symmetric product $S^d(C).$  This is smooth, and the proof is done via local coordinates, as follows.
1) In the case where $C={\mathbb A}^1$, we are still in the affine case.  We know by the theorem on elementary symmetric polynomials (Waring) that (here $X=(X_i)_{i\leq n}$) 
$$k[X]^{S_n}=k[s_1, \cdots , s_n],$$
where $s_i$ are defined by 
$$T^n-s_1T^{n-1} + \cdots +(-1)^ns_n= \prod (T-X_i).$$
2) In the case of a smooth curve $C$, consider a point  $\sum n_i P_i$, where $P_i \in C$ and $\sum n_i=d.$  Without much effort (but by proceeding carefully) one may reduce to the above by choosing local coordinates and considering the decomposition group of a point of $C^d$ above our chosen point in $S^dC$.


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*If $X$ is a smooth projective surface, then the desingularisation of $S^dX$ (which is always non-smooth for $\dim X\geq 2$ as the branch locus is a union of diagonals which has codimension $\leq 2$) has a correspondence with the Hilbert scheme of zero-dimensional subschemes of $X$ of length $d$.  The standard reference is H. Nakajima's book:


http://books.google.com.br/books/about/Lectures_on_Hilbert_Schemes_of_Points_on.html?id=pjjgNhDVgcgC&redir_esc=y


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*I remember that I.G. Macdonald has a book on symmetric functions, which perhaps is of some use to you, though it won't be the answer you seek, as your question stands.

*There is a paper by E. Freitag on J. Crelle on actions of finite groups on varieties (it contains some results on numerical invariants in the case of complex smooth projective varieties).
http://www.researchgate.net/publication/243110046_Die_Struktur_der_Funktionenkrper_zu_hyperabelschen_Gruppen._II
Finally, maybe you have heard of the Molien series (related to finite group actions on projective schemes over a field).
