On Euler's abelian relation of the web induced by the forgetful maps on the moduli space ℳ0,n+3: the case when n is even
Abstract
In a previous paper bf [P1], for every n≥ 2, we studied the curvilinear web 𝒲0,n+3 on the moduli space ℳ0,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 G2or(Rn+3) of oriented $2$-planes in Rn+3. In this note, we obtain an explicit antisymmetrized formula for Euler's abelian relation for every even n≥ 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.
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BibTeX
@misc{pirio2026eulers,
title = {{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}},
author = {Luc Pirio},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28956},
eprint = {2610.28956},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28956},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Euler's abelian relation of the web induced by the forgetful maps on the moduli space ℳ_0,n+3: the case when n is even
AU - Luc Pirio
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28956
UR - https://arxiv.org/abs/2610.28956
AB - In a previous paper bf [P1], for every n≥ 2, we studied the curvilinear web 𝒲_0,n+3 on the moduli space 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^or(R^n+3) of oriented $2$-planes in R^n+3. In this note, we obtain an explicit antisymmetrized formula for Euler's abelian relation for every even n≥ 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.
ER -