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

TL;DR

This paper studies the geometric and algebraic properties of certain webs on moduli spaces of points on the projective line, revealing their maximal rank, symmetry representations, and explicit abelian relations, including a conjectural formula for a special relation.

Contribution

It provides a detailed analysis of the abelian relations of $(n+3)$-webs on $ ext{M}_{0,n+3}$, correcting previous approaches and establishing their maximal rank and symmetry properties.

Findings

The web $oldsymbol{ ext{W}}_{0,6}$ has maximal rank and rational ARs forming an irreducible $rak{S}_6$-module.

The web $oldsymbol{ ext{W}}_{0,n+3}$ has maximal rank for all $n extgreater{}=2$.

A conjectural explicit formula for the Euler's abelian relation $oldsymbol{ ext{E}}_n$ is proposed and verified for $n extless{}=12$.

Abstract

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 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 $\boldsymbol{\mathcal W}_{{0,6}}$. We show that this web is isomorphic to the web formed by the lines contained in Segre's cubic primal $\boldsymbol{S}\subset \mathbf P^4$ 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 $\boldsymbol{\mathcal E}_n$', 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 theory of polylogarithmic forms. We give an explicit conjectural formula for the components of $\boldsymbol{\mathcal E}_n$ that we prove to be correct for $n\leq 12$.