On the $(n+3)$-webs by rational curves induced by the forgetful maps on the moduli spaces ${\mathcal{M}}_{0,n+3}$

We discuss the curvilinear web ${\mathcal{W}}_{0,n+3}$ on the moduli space ${\mathcal{M}}_{0,n+3}$ of projective configurations of $n+3$ points on ${\mathbf{P}}^1$ defined by the $n+3$ forgetful maps $\mathcal M_{0,n+3}\rightarrow \mathcal M_{0,n+2}$. We recall classical results which show that this web is linearizable when $n$ is odd, or is equivalent to a web by conics when $n$ is even. We then turn to the abelian relations (ARs) of these webs. After recalling the well-known case when $n=2$ (related to the 5-terms functional identity of the dilogarithm), we focus on the case of the 6-web ${\mathcal{W}}_{0,6}$. We show that this web is isomorphic to the web formed by the lines contained in Segre’s cubic primal $\mathit{S}\subset {\mathbf{P}}^4$ and that a kind of Abel's theorem' allows to describe the ARs of $\boldsymbol{\mathcal W}_{{0,6}}$ by means of the abelian 2-forms on the Fano surface $F_1(\boldsymbol{S})\subset G_1(\mathbf P^4)$ of lines contained in $\boldsymbol{S}$. We deduce from this that $\boldsymbol{\mathcal W}_{{0,6}}$ has maximal rank with all its ARs rational, and that these span a space which is an irreducible $\mathfrak S_6$-module. Then we take up an approach due to Damiano that we correct in the case when $n$ is odd: it leads to an abstract description of the space of ARs of $\boldsymbol{\mathcal W}_{0,n+3}$ as a $\mathfrak S_{n+3}$-representation. In particular, we obtain that this web has maximal rank for any $n\geq 2$. Finally, we consider Euler’s abelian relation ${\mathcal{E}}_n$’, a particular AR for ${\mathcal{W}}_{0,n+3}$ constructed by Damiano from a characteristic class on the grassmannian of 2-planes in ${\mathbf{R}}^{n+3}$ by means of Gelfand-MacPherson theory of polylogarithmic forms. We give an explicit conjectural formula for the components of ${\mathcal{E}}_n$ that we prove to be correct for $n\le 12$.