TITLE: Koszulness of the cohomology ring of moduli of stable genus zero curves
QUESTION [14 upvotes]: 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?

REPLY [11 votes]: 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.