# 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.

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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 mathoverflow.net/questions/1558/… 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: