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If $X$ is a space, the configuration space of $n$ (distinct) points in $X$ is $C_n(X)=F_n(X)/\Sigma_n$, where $F_n(X) = \{x \in X^n : \forall i,j (i \neq j \Rightarrow x_i \neq x_j)\}$ is the configuration space of $n$ distinct ordered points.

I would like to make such a construction if $X$ is a scheme. Here is my naive idea: Let's assume that $X$ is separated (i.e. the diagonal $\Delta : X \to X \times X$ is closed), hence $U:=(X \times X) \setminus \Delta(X) \hookrightarrow X \times X$ is open. For all $i \neq j$ we have a morphism $(p_i,p_j) : X^n \to X \times X$. The preimage of $U$ is an open subscheme of $X^n$. Let $F_n(X)$ be the intersection of all these open subschemes. The action of $\Sigma_n$ on $X^n$ restricts to an action on $F_n(X)$.

Let us define the scheme $C_n(X):=F_n(X)/\Sigma_n$ provided that this scheme quotient exists. For example, there is a general result showing that this is the case when $X$ is quasi-projective. (Or does this specific quotient always exist?)

Now my question is: How can we describe the functor of points of $C_n(X)$? That is, if $T$ is a scheme, how can we describe $\hom(T,C_n(X))$ in terms of $\hom(T,X)$?

Then $\hom(T,F_n(X))$ is the set of those $f \in \hom(T,X)^n$ which are not only pairwise distinct, but rather "disjoint", i.e. for $i \neq j$ we have $\mathrm{eq}(f_i,f_j)=\emptyset$, where $\mathrm{eq}$ denotes the equalizer in the category of schemes. But how to describe morphisms into a quotient? There is a natural map $$\hom(T,F_n(X))/\Sigma_n \to \hom(T,F_n(X)/\Sigma_n),$$ but it doesn't seem to be an isomorphism. In fact, the left hand side doesn't seem to be a sheaf in $X$. So perhaps $\hom(-,C_n(X))$ is the sheaf associated to the presheaf $\hom(T,F_n(-))/\Sigma_n$?

Also, is anything known about quasi-coherent sheaves on $C_n(X)$?

I am also interested in variants of this construction. For example, we may also allow equal points and consider $X^n / \Sigma_n$.

Martin Brandenburg
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  • Is $\Delta$ the diagonal mapping? –  May 26 '14 at 21:52
  • Indeed, $\Delta$ is the diagonal. – Martin Brandenburg May 26 '14 at 22:05
  • if $X=\mathbb C\backslash{0}$ and $f:X\to C_n(X)$ is given by $x\mapsto{n\text{-th roots of }x}$ then $f$ can't be described as an $n$-tuple of maps $X\to X$. – user8268 May 26 '14 at 22:09
  • Describing maps into a quotient is difficult when quotients are not stable under pullback; but when quotients are stable under pullback, you can describe morphisms into the quotient à la principal bundles and classifying spaces. This applies in the category of sheaves, but I don't see any reason why it should apply in the category of schemes... – Zhen Lin May 26 '14 at 22:47
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    So, basically, you're looking for a functorial description of the open sub scheme of the Hilbert scheme of $n$ points parametrising distinct points, right? – Ben May 27 '14 at 08:59
  • @user8268: Why is $f$ a morphism of schemes? @ Ben A.: I'm not very familiar with Hilbert schemes, so I cannot exclude that their theory may be helpful here. – Martin Brandenburg May 28 '14 at 08:47
  • $C_n(X)$ (for my $X$) is the space of monic polynomials of degree $n$, $y^n+a_{n-1}y^{n-1}+\dots +a_0$, without multiple roots and without $0$ as a root, i.e. it is $\mathbb C^n$ ($a_k$'s are the coordinates) minus the discriminant variety and minus the hyperplane $a_0=0$. Now $f$ is given by $a_0=x$, $a_k=0$ for $k>0$. – user8268 May 28 '14 at 08:53
  • Even if the answer to my question was yes, this wouldn't be helpful directly, because in general, pretty much nothing is known about the open subset of distinct points. Its closure, the so called good component however can in certain cases be constructed as a blow up of the symmetric power. Unfortunately, I haven't read this article very extensively, so I can neither provide more details right now nor can I assure that it would be helpful to your question. – Ben May 28 '14 at 09:55

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