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We study and compare the webs $\boldsymbol{\mathcal W}_{{\rm dP}_d}$ definedby the conic fibrations on a given smooth del Pezzo surface ${\rm dP}_d$ ofdegree $d$ for $d=4$ and $d=5$. In a previous paper, we proved that for anypositive $d\leq 6$, the web by conics $\boldsymbol{\mathcal W}_{{\rm dP}_d}$carries a particular abelian relation ${\bf HLog}_d$, whose components all areweight $7-d$ antisymmetric hyperlogarithms. The web $\boldsymbol{\mathcalW}_{{\rm dP}_5}$ is a geometric model of the exceptional Bol's web and therelation ${\bf HLog}_5$ 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 differentkinds. We show that almost all of them admit natural generalizations to thepair $\big( \boldsymbol{\mathcal W}_{{\rm dP}_4}, {\bf HLog}_4\big)$.

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