On Euler's abelian relation of the web induced by the forgetful maps on the moduli space $\boldsymbol{\mathcal M_{0,n+3}}$: the case when $n$ is even

Abstract

In a previous paper ${\bf [P1]}$, for every $n\geq 2$, we studied the curvilinear web $\boldsymbol{\mathcal W}_{0,n+3}$ on the moduli space $\boldsymbol{\mathcal M}_{0,n+3}$, defined by the $n+3$ forgetful maps $ \boldsymbol{\mathcal M}_{0,n+3}\rightarrow \boldsymbol{\mathcal M}_{0,n+2}$, with particular emphasis on its abelian relations. In particular, when $n$ is even, we conjectured an explicit formula for its Euler abelian relation. This relation is constructed à la Gelfand-MacPherson from an invariant differential form representing the appropriate power of the Euler class of the tautological bundle of the Grassmannian manifold $G_2^{\mathrm{or}}(\mathbf R^{n+3})$ of oriented $2$-planes in $\mathbf R^{n+3}$. In this note, we obtain an explicit antisymmetrized formula for Euler's abelian relation for every even $n\geq 2$, and prove that it coincides, up to the explicit factor $-(n+1)(n+2)/2$, with the formula conjectured in ${\bf [P1]}$, which is therefore established for all even $n$.