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

Abstract

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 ${\mathcal{M}}_{0,n+3}\to {\mathcal{M}}_{0,n+2}$. 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 $\mathbf{S}\subset {\mathbf{P}}^4$ and that a kind of `Abel's theorem' allows to describe the ARs of ${\mathcal{W}}_{0,6}$ by means of the abelian 2-forms on the Fano surface $F_1(\mathbf{S})\subset G_1({\mathbf{P}}^4)$ of lines contained in $\mathbf{S}$. We deduce from this that ${\mathcal{W}}_{0,6}$ 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 ${\mathcal{W}}_{0,n+3}$ as a ${\mathfrak{S}}_{n+3}$-representation. In particular, we obtain that this web has maximal rank for any $n\ge 2$. 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 $n\le 12$.

arXiv comment121 pages, 4 figures. Second version : a few typos have been corrected