Let $n \geq 3$. The ring $H^\bullet(\overline{M}\_{0,n},\mathbf Q)$ was determined by Sean Keel. It is generated by the cohomology classes of boundary divisors $D\_{A,B}$ corresponding to partitions $A \sqcup B = \{1,\ldots,n\}$ of the marked points with $|A|, |B| \geq 2$, and where $D\_{A,B} = D\_{B,A}$. All relations are given by: (i) demanding that the product $D\_{A,B} \cdot D\_{A',B'}$ vanishes if the two divisors are disjoint (i.e. if there are no containments between the four sets $A,A',B$ and $B'$); (ii) the relation $$ \sum\_{\substack{\{i,j\} \subseteq A \\ \{k,l\} \subseteq B}} D\_{A,B} = \sum\_{\substack{\{i,k\} \subseteq A \\ \{j,l\} \subseteq B}} D\_{A,B} $$ which follows by pulling back the WDVV relation on $\overline{M}\_{0,4}$ to $\overline M\_{0,n}$. 


It follows in particular that $H^{2\bullet}(\overline{M}\_{0,n},\mathbf Q)$ is a quadratic algebra. I was asked during a seminar today whether this algebra is Koszul, but I had no idea what to answer. So, is it Koszul? If so, is its Koszul dual interesting?



Koszulness of the cohomology ring of moduli of stable genus zero curves
It is: <https://arxiv.org/abs/1902.06318> - this paper also explains how to use the Koszul dual algebra for something, where something is estimating Betti numbers of the free loop spaces of $\overline{M}\_{0,n}$; those, by Gromov & Ballmann-Ziller, allow one to estimate the number of closed geodesics of bounded length.


