On the $(n+3)$-webs by rational curves induced by the forgetful maps on the moduli spaces $\mathcal M_{0,n+3}$
Volume 584 / 2023
Abstract
For $n\geq 2$, we discuss the curvilinear web $\boldsymbol{\mathcal W}_{0,n+3}$ 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 (first obtained by Room) which show that this web is linearizable when $n$ is odd, and 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 classical and well-known case when $n=2$ (related to the famous $5$-term functional identity of the dilogarithm), we focus on the case of the 6-web $\boldsymbol{\mathcal W}_{{0,6}}$. We show that this web is isomorphic to the web formed by the projective lines contained in Segre’s cubic primal $\boldsymbol{S}\subset \mathbf P^4$. Using this together with a kind of ‘Abel’s theorem’, we 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 AR 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 an $\mathfrak S_{n+3}$-representation. In particular, we find that this web has maximal rank for any $n\geq 2$. Finally, we consider ‘Euler’s abelian relation $\boldsymbol{\mathcal E}_n$’, a particular AR for $\boldsymbol{\mathcal W}_{0,n+3}$ constructed by Damiano from a characteristic class on the Grassmannian of 2-planes in $\mathbf R^{n+3}$ by means of Gelfand–MacPherson’s theory of polylogarithmic forms. We give an explicit conjectural formula for the components of $\boldsymbol{\mathcal E}_n$, which involves only rational (resp. rational and logarithmic) terms for $n$ odd (resp. for $n$ even). By means of direct computations, we prove that our explicit formulas are indeed correct for $n\leq 12$.
