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 $X^n / \Sigma_n$.