A story of webs: the webs by conics on del Pezzo quartic surfaces and Gelfand-MacPherson's web of the spinor tenfold

Abstract

In a previous paper, we studied the web by conics $\boldsymbol{\mathcal W}_{{\rm dP}_4}$ on a del Pezzo quartic surface ${\rm dP}_4$ and proved that it enjoys suitable versions of most of the remarkable properties satisfied by Bol's web $\boldsymbol{\mathcal B}$. In particular, Bol's web can be seen as the toric quotient of the Gelfand-MacPherson web naturally defined on the $A_4$-grassmannian variety $G_2(\mathbf C^5)$ and we have shown that $\boldsymbol{\mathcal W}_{{\rm dP}_4}$ can be obtained in a similar way from the web $\boldsymbol{\mathcal W}^{GM}_{ \hspace{-0.05cm} \boldsymbol{\mathcal Y}_5}$ which is the quotient by the Cartan torus of ${\rm Spin}_{10}(\mathbf C)$, of the Gelfand-MacPherson 10-web naturally defined on the tenfold spinor variety $\mathbb S_5$, a peculiar projective homogenous variety of type $D_5$. In the present paper, by means of direct and explicit computations, we show that many of the remarkable similarities between $\boldsymbol{\mathcal B}$ and $\boldsymbol{\mathcal W}_{{\rm dP}_4}$ actually can be extended to, or from an opposite perspective, can be seen as coming from some similarities between Bol's web and $\boldsymbol{\mathcal W}^{GM}_{ \hspace{-0.05cm} \boldsymbol{\mathcal Y}_5}$. The latter web can be seen as a natural uniquely defined rank 5 generalization of Bol's web. In particular, it carries a peculiar 2-abelian relation, denoted by ${\bf HLOG}_{ \boldsymbol{\mathcal Y}_5}$, which appears as a natural generalization of Abel's five terms relation of the dilogarithm and from which one can recover the weight 3 hyperlogarithmic functional identity of any quartic del Pezzo surface.