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 ${\mathcal{W}}_{{\mathrm{dP}}_4}$ on a del Pezzo quartic surface ${\mathrm{dP}}_4$ and proved that it enjoys suitable versions of most of the remarkable properties satisfied by Bol's web $\mathcal{B}$. 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 ${\mathcal{W}}_{{\mathrm{dP}}_4}$ can be obtained in a similar way from the web ${\mathcal{W}}_{{\mathcal{Y}}_5}^{GM}$ which is the quotient by the Cartan torus of ${\mathrm{Spin}}_{10}(\mathbf{C})$, 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 $\mathcal{B}$ and ${\mathcal{W}}_{{\mathrm{dP}}_4}$ actually can be extended to, or from an opposite perspective, can be seen as coming from some similarities between Bol's web and ${\mathcal{W}}_{{\mathcal{Y}}_5}^{GM}$. 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 ${\mathbf{HLOG}}_{{\mathcal{Y}}_5}$, 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.

73 pages, a few figures. Preliminary version, comments are welcome