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

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