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In this question, I am working over $\mathbb{C}$ and with rational Chow groups.

I am interested in the state of the art for vanishing theorems of the form $A^{k}(\mathcal{M}_{g,n})=0$ for some choies of $k,g,n$. (Note that I have restricted to the space of smooth curves.) One specific question I am especially interested in is: for which values of $k$ is $A^k(\mathcal{M}_{1,n})=0$?

Some example results in this direction: due to Mumford ($g=2$), Faber ($g=3,4$), Izadi ($g=5$), and Penev-Vakil ($g=6$), we know that all classes in $A^k(\mathcal{M}_g)$ are tautological, and hence by Looijenga ("On the tautological ring of $\mathcal{M}_g$") we have $A^{k}(\mathcal{M}_g)=0$ for $k\ge g-1$. It is also shown along the way to some of these calculations that $A^{k}(\mathcal{M}_{g,n})=0$ for certain values of $k$ when $g,n$ are small. For arbitrary $n$, Keel has computed the full Chow ring of $\overline{M}_{0,n}$; I think it follows that all Chow groups of $M_{0,n}$ vanish.

Additionally, if there is a good summary in the literature of what is known for singular cohomology too, I would love to know where to find it.

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    $\begingroup$ For $\mathcal M_{1,n}$ I would think the singular cohomology is basically known in terms of modular forms, but maybe not if the spectral sequence is tricky. $\endgroup$
    – Will Sawin
    Jan 13, 2020 at 20:12
  • $\begingroup$ Can you clarify what you mean by "the spectral sequence"? $\endgroup$
    – Hans Sachs
    Jan 14, 2020 at 0:14
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    $\begingroup$ I guess I mean iteratively applying the Leray spectral sequence to the fibrations $\mathcal M_{1,n} \to \mathcal M_{1,n-1}$, but there are a few different geometric strategies one could try. $\endgroup$
    – Will Sawin
    Jan 14, 2020 at 0:58

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