Motive of unlabeled fulton-macpherson configuration space?

Question

I am working on the class of the Fulton-Macpherson compactification of configuration space in the Grothendieck ring of varieties, over the field of complex numbers. As a first step, I am wondering about the case of 2 points, i.e. the motive of $X[2]/S_2$ for a nonsingular variety $X$. For the case the two points coincide, one was lead naturally to the space $(\mathbb{P}(T_x^2/T_x))/S_2\cong\mathbb{P}((\mathbb{A}^n)^2/\mathbb{A}^n)$. My question is: (1) does this space admit an expression in terms of the Lefschetz motive $\mathbb{L}$? (2) does this space admit a cell decomposition? (i.e. is it a disjoint union of locally closed subvarieties isomorphic to affine spaces?)

Answer 1

If we identify $T_x^2 / T_x$ with $\mathbb A^n$, then the action of $S_2$ is multiplication by $-1$. So $S_2$ acts trivially on the projectivization and you are just asking about the existence of a cell decomposition of projective space.

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A useful reference might be the last parts of Ezra Getzler's old preprint "Mixed Hodge structures of configuration spaces". He derives explicit formulas for the $S_n$-equivariant "motivic" Euler characteristic of $X[n]$ in terms of the motivic Euler characteristic of $X$. Here "motivic" means that the Euler characteristic is taken in a suitable ring of the form $K$ tensored with the representation ring of $S_n$, where $K$ is e.g. the Grothendieck group of $\ell$-adic Galois representations or the Grothendieck group of Hodge structures. So his formulas include in particular the equivariant Hodge-Deligne polynomial. Then taking $S_n$-invariants just corresponds to the substitution $p_n \mapsto x^n$, where $p_n$ is the $n$th power sum and $x$ is an indeterminate.