On the 10-web by conics on the quartic del Pezzo surface

Abstract

We study and compare the webs ${\mathcal{W}}_{{\mathrm{dP}}_d}$ defined by the conic fibrations on a given smooth del Pezzo surface ${\mathrm{dP}}_d$ of degree $d$ for $d=4$ and $d=5$. In a previous paper, we proved that for any positive $d\le 6$, the web by conics ${\mathcal{W}}_{{\mathrm{dP}}_d}$ carries a particular abelian relation ${\mathbf{HLog}}_d$, whose components all are weight $7-d$ antisymmetric hyperlogarithms. The web ${\mathcal{W}}_{{\mathrm{dP}}_5}$ is a geometric model of the exceptional Bol's web and the relation ${\mathbf{HLog}}_5$ corresponds to the famous `Abel's identity' $(\mathcal{A}\mathbf{b})$ of the dilogarithm. Bol's web together with $(\mathcal{A}\mathbf{b})$ enjoy several remarkable properties of different kinds. We show that almost all of them admit natural generalizations to the pair $({\mathcal{W}}_{{\mathrm{dP}}_4},{\mathbf{HLog}}_4)$.

100 pages

References in corpus (4)

• On the Cox ring of Del Pezzo surfaces
• Clean Single-Valued Polylogarithms
• Pentagon relation in quantum cluster scattering diagrams
• Initial degenerations of spinor varieties