Two-dimensional cycle classes on \(\overline{\mathcal{M}}_{0,n}\) (Q2152852)

The paper studies the cohomology -- or equivalently in this case the Chow groups -- of moduli spaces of stable curves of genus 0 with n markings \(\overline{\mathcal{M}}_{0,n}\) with rational coefficient. As a consequnece of the work of \textit{A.-M. Castravet} and \textit{J. Tevelev} [Algebr. Geom. 7, No. 6, 722--757 (2020; Zbl 1467.14069)], the full cohomology ring \(H^*(\overline{\mathcal{M}}_{0,n}, \mathbb{Q})\) is a premutation representation of the symmetric group \(S_n\), i.e. a representation of \(S_n\) with a basis \(B\) such that the action of \(S_n\) on the vector space restricts to an action on \(B\). Then, the general open problem motivating the present study can be formulated as: \textit{is \(H^{2k}(\overline{\mathcal{M}}_{0,n}, \mathbb{Q})\) a permutation representation of \(S_n\) for all \(k\)?} \medskip This question had been answered positively by \textit{G. Farkas} and \textit{A. Gibney} if \(k=1\) or \(n-1\) [Trans. Am. Math. Soc. 355, No. 3, 1183--1199 (2003; Zbl 1039.14008)], and the present paper solves this problem for \(k=2\) or \(n-2\). Their proof relies on the construction of a decomposition of each cohomology group: \[ H^{2k}(\overline{\mathcal{M}}_{0,n}, \mathbb{Q})=\bigoplus_{r=1}^{\min(k,n-3-k)} Q_{k,n}^r. \] To construct this decomposition, the authors use the fact that the cohomology of the moduli spaces of curves of genus 0 is spanned by the classes associated to the boundary components which are encoded in the so-called stable graphs (see [\textit{S. Keel}, Trans. Am. Math. Soc. 330, No. 2, 545--574 (1992; Zbl 0768.14002)]). Then, they construct a partition of the set of stable graphs (according to the valencies of the vertices) which in turn provides a decomposition of the cohomolgy groups by considering the span of each subset of this partition. The authors prove that \(Q_{k,n}^1\) and \(Q_{k,n}^2\) are permutation representation of \(S_n\) for all \(k\) and \(n\) which provides the main result of the paper. Unfortunately, the space \(Q_{3,9}^3\) is not a permutation representation of \(S_9\) in general, thus the strategy cannot be pushed further to address the open problem for general \(k\) (yet, the authors checked that \(H^6(\overline{\mathcal{M}}_{0,9}, \mathbb{Q})\) is still a permutation representation, thus the general problem remains open).