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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?

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  • $\begingroup$ This definition doesn't define a finite cover of $\mathcal M_{0,n}$ over $\mathbb Q$ - you need to pick a base point. However, it's possible to check, regardless of the base point, that the cohomology of $\mathcal M_{0,4}^s$ is Artin-Tate, by just bashing it out. $\endgroup$
    – Will Sawin
    Commented Jun 25, 2020 at 16:39
  • $\begingroup$ Thank you! I guess that I was implicitly working over C. $\endgroup$
    – Daniil Rudenko
    Commented Jun 25, 2020 at 16:41
  • $\begingroup$ You want the Hodge structure to be mixed Tate type then? It is for $n=4$, because $\mathcal M_{0,4}^s $ is $ \mathbb P^1$ minus six points. Probably is for $n=5$ as well. I guess this should stop at some point though, but I don't know. $\endgroup$
    – Will Sawin
    Commented Jun 25, 2020 at 17:00
  • $\begingroup$ I see, thank you. $\endgroup$
    – Daniil Rudenko
    Commented Jun 25, 2020 at 17:01

1 Answer 1

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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 $n\geq 12$.

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  • $\begingroup$ Thank you, this is very clarifying! $\endgroup$
    – Daniil Rudenko
    Commented Jun 25, 2020 at 17:29
  • $\begingroup$ Also, I would be really glad for a reference, in what sense it is a spin moduli space? I found a lot of contradicting definitions in the literature. I guess that it should be the underlying variety for the corresponding super-moduli space, but even that is not completely clear to me. $\endgroup$
    – Daniil Rudenko
    Commented Jul 1, 2020 at 18:54
  • $\begingroup$ @DaniilRudenko I know nothing about the spin terminology unfortunately. $\endgroup$
    – Will Sawin
    Commented Jul 1, 2020 at 20:02
  • $\begingroup$ Thank you, I will try to find out. $\endgroup$
    – Daniil Rudenko
    Commented Jul 1, 2020 at 20:03

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