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We discuss the curvilinear web $\boldsymbol{\mathcal W}_{0,n+3}$ on themoduli space $\mathcal M_{0,n+3}$ of projective configurations of $n+3$ pointson $\mathbf P^1$ defined by the $n+3$ forgetful maps $\mathcalM_{0,n+3}\rightarrow \mathcal M_{0,n+2}$. We recall classical results whichshow that this web is linearizable when $n$ is odd, or is equivalent to a webby conics when $n$ is even. We then turn to the abelian relations (ARs) ofthese webs. After recalling the well-known case when $n=2$ (related to the5-terms functional identity of the dilogarithm), we focus on the case of the6-web $\boldsymbol{\mathcal W}_{{0,6}}$. We show that this web is isomorphic tothe 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' allowsto describe the ARs of $\boldsymbol{\mathcal W}_{{0,6}}$ by means of theabelian 2-forms on the Fano surface $F_1(\boldsymbol{S})\subset G_1(\mathbfP^4)$ 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. Inparticular, we obtain that this web has maximal rank for any $n\geq 2$.Finally, we consider `Euler's abelian relation $\boldsymbol{\mathcal E}_n$', aparticular AR for $\boldsymbol{\mathcal W}_{0,n+3}$ constructed by Damiano froma characteristic class on the grassmannian of 2-planes in $\mathbf R^{n+3}$ bymeans of Gelfand-MacPherson theory of polylogarithmic forms. We give anexplicit conjectural formula for the components of $\boldsymbol{\mathcal E}_n$that we prove to be correct for $n\leq 12$.

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