Let $\overline{M}_{0,n}$ be the moduli space of genus zero curves with $n$ marked points. Let $I=\{\{S,S^c\}|S\subset\{1,\dots,n\},|S|\geq2, |S^c|\geq2\}$ be the set of partitions of $\{1,\dots n\}$ into two subsets, each has at least two elements.

Keel shown that the cohomology group $H^2(\overline{M}_{0,n},\mathbb{Z})$ is generated by boundary divisor classes $\delta_{\{S,S^c\}}$, where $\{S,S^c\}\in I$. (Here $\delta_S$ is the divisor in $\overline{M}_{0,n}$, consisting of curves with two components, marked by $S$ and $S^c$, and their further degenerations.)

Thus we have a presentation $$\mathrm0\to K\to\bigoplus_{\{S,S^c\}\in I}\mathbb{Z}\cdot\delta_{\{S,S^c\}}\to H^2(\overline{M}_{0,n},\mathbb{Z})\to 0.$$

Keel shown that the kernel $K$ is generated by equations $$\sum_{i,j\in S;k,l\notin S}\delta_{\{S,S^c\}}=\sum_{i,k\in S;j,l\notin S}\delta_{\{S,S^c\}}=\sum_{i,l\in S;k,j\notin S}\delta_{\{S,S^c\}},$$ for any four distinct elements $\{i,j,k,l\}\subset\{1,\dots,n\}$. These give $2\cdot{n\choose 4}$ such equations. But the rank of $K$ is only $\frac{n(n-3)}{2}$. ($\#I=2^{n-1}-n-1$, $\mathrm{rank}H^2(\overline{M}_{o,n},\mathbb{Z})=2^{n-1}-{n\choose2}-1$), so these relations are very much dependent..

The question is, would there be a good presentation of $K$, my goal is to calculate the group cohomology $$H^1(S_n,H^2(\overline{M}_{0,n},\mathbb{Z}))?$$


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