2025/07/16 by Pirio, Luc
#11G55 #14M15 #53A60 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.12180
In a previous paper, we studied the web by conics \boldsymbol\mathcal W_\rm dP4 on a del Pezzo quartic surface \rm dP4 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 A4-grassmannian variety G2(\mathbf C5) and we have shown that \boldsymbol\mathcal W_\rm dP4 can be obtained in a similar way from the web \boldsymbol\mathcal WGM_ \hspace-0.05cm \boldsymbol\mathcal Y5 which is the quotient by the Cartan torus of \rm Spin10(\mathbf C), of the Gelfand-MacPherson 10-web naturally defined on the tenfold spinor variety \mathbb S5, a peculiar projective homogenous variety of type D5. 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 dP4 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 WGM_ \hspace-0.05cm \boldsymbol\mathcal Y5. 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 Y5, 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.