Cohomology of the moduli space of rational curves with $n$ marked points with spin structure Consider $\mathcal{M}_{0,n}$- the moduli space of rational curves with $n$ marked points. A map
$$
p\colon \pi_1(\mathcal{M}_{0,n})\longrightarrow H_1(\mathcal{M}_{0,n},\mathbb{Z}/2\mathbb{Z})
$$
defines a finite cover  $\mathcal{M}_{0,n}^s.$ One can think of it as a space, on which square roots of cross ratios are regular functions. If I understand correctly, this is the underlying manifold of the super-moduli space of  rational curves with $n$ marked points of NS type.
Question 1 (easy): Points of $\mathcal{M}_{0,n}^s$ should parametrize marked rational curves with spin structure, but I am not sure that I understand, what that means. I would be grateful for an  explanation or a reference.
Question 2 (more interesting): What is known about cohomology of $\mathcal{M}_{0,n}^s?$ For instance, are they mixed Tate?
 A: The cohomology is not mixed Tate for $n\geq 12$, and this possibly can be improved.
We can view $\mathcal M_{0,n}$ as the locus of $n$-tuples of points $x_1,\dots,x_n$ in $\mathbb P^1$ which are all distinct, and where $x_1=0, x_2=1, x_3=\infty$.
Consider the $2^{n-4}$-fold cover of $\mathcal M_{0,n}$ defined by adjoining square-roots of $x_j (x_j-1) (x_j- x_4)$ for $j$ from $5$ to $n$. This cover is an open subset of the $n-4$-fold self-product of the Legendre family of elliptic curves $y^2 =x (x-1) (x-x_4)$.
This $n-4$-fold self-product includes in its cohomology the space of cusp forms of weight $(n-4)+2$ and level $2$. The associated Hodge structures are pure of weight $(n-4)+1$, lie in degree $(n-4)+1$, and are not mixed Tate (unless they are trivial). Because of this purity they are preserved when restricting to an open subset, and thus these non-mixed-Tate cohomology classes show up in this $2^{n-4}$-fold covering.
Because the spin moduli space is the universal covering with monodromy group $\mathbb F_2^k$, it admits a finite map to this $2^{n-4}$-fold covering, and so the non-mixed-Tate classes will also show up there.
There is a cusp form of level $2$ and weight $8$ so this will show up for $n\geq 12$.
