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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.

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Do you know how to construct products of schemes? Do you know how to quotient schemes by the action of a finite group? – Qiaochu Yuan Aug 3 '12 at 17:09
I didn't know that you could quotient schemes by the action of a finite group. How would you do that? – only Aug 3 '12 at 17:21
If $R$ is a commutative ring, then an action of $G$ on $\text{Spec } R$ by scheme automorphisms is equivalent to an action of $G$ on $R$ by ring automorphisms, and $\text{Spec } R^G$ is a sensible model of the quotient scheme (where $R^G$ denotes the invariant subring of $R$). The inclusion $R^G \to R$ dualizes to the quotient map $\text{Spec } R \to \text{Spec } R^G$. To extend this to schemes it suffices to find a covering by $G$-invariant affine opens; unfortunately I am not sure if this is always possible... – Qiaochu Yuan Aug 3 '12 at 17:25
Hmm, the answers to… seem to suggest that it's not always possible to quotient by a finite group... – only Aug 3 '12 at 18:17
Yes, okay, the issue is that orbits may not be contained in affine opens. For any scheme $S$ such that a finite subset of $S$ is contained in an affine open this is not a problem for the action of the symmetric group on $S \times S \times ... \times S$. – Qiaochu Yuan Aug 3 '12 at 18:25

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.

  • 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.

  • 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 $S^dC$.

  • 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:

  • 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).

Finally, maybe you have heard of the Molien series (related to finite group actions on projective schemes over a field).

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