On the (n + 3)-webs by rational curves induced by the forgetful maps on the moduli spaces M_0,n+3

For n ≥ 2, we discuss the curvilinear web W_0,n+3 on the moduli space M_0,n+3 of projective configurations of n + 3 points on P¹, defined by the n + 3 forgetful maps M_0,n+3 → 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 W_0,6. We show that this web is isomorphic to the web formed by the projective lines contained in Segre’s cubic primal S ⊂ P⁴. Using this together with a kind of ‘Abel’s theorem’, we describe the ARs of W_0,6 by means of the abelian 2-forms on the Fano surface F₁(S) ⊂ G₁(P⁴ ) of lines contained in S. We deduce from this that W_0,6 has maximal rank with all its AR rational, and that these span a space which is an irreducible S₆-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 W_0,n+3 as an S_n+3-representation. In particular, we find that this web has maximal rank for any n ≥ 2. Finally, we consider ‘Euler’s abelian relation E_n’, a particular AR for W_0,n+3 constructed by Damiano from a characteristic class on the Grassmannian of 2-planes in Rⁿ⁺³ by means of Gelfand–MacPherson’s theory of polylogarithmic forms. We give an explicit conjectural formula for the components of 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 ≤ 12.