2022/04/10 by Pirio, Luc
#(11G55 #14J) #53A60 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2204.04772
We discuss the curvilinear web \boldsymbol\mathcal W0,n+3 on the moduli space \mathcal M0,n+3 of projective configurations of n+3 points on \mathbf P1 defined by the n+3 forgetful maps \mathcal M0,n+3→ \mathcal M0,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 W0,6. We show that this web is isomorphic to the web formed by the lines contained in Segre's cubic primal \boldsymbolS⊂ \mathbf P4 and that a kind of `Abel's theorem' allows to describe the ARs of \boldsymbol\mathcal W0,6 by means of the abelian 2-forms on the Fano surface F1(\boldsymbolS)⊂ G1(\mathbf P4) of lines contained in \boldsymbolS. We deduce from this that \boldsymbol\mathcal W0,6 has maximal rank with all its ARs rational, and that these span a space which is an irreducible \mathfrak 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 \boldsymbol\mathcal W0,n+3 as a \mathfrak Sn+3-representation. In particular, we obtain that this web has maximal rank for any n≥ 2. Finally, we consider `Euler's abelian relation \boldsymbol\mathcal En', a particular AR for \boldsymbol\mathcal W0,n+3 constructed by Damiano from a characteristic class on the grassmannian of 2-planes in \mathbf Rn+3 by means of Gelfand-MacPherson theory of polylogarithmic forms. We give an explicit conjectural formula for the components of \boldsymbol\mathcal En that we prove to be correct for n≤ 12.