On the 10-web by conics on the quartic del Pezzo surface
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910320597925888 |
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| author | Pirio, Luc |
| author_facet | Pirio, Luc |
| contents | 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 $\big( \boldsymbol{\mathcal W}_{{\rm dP}_4}, {\bf HLog}_4\big)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06711 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the 10-web by conics on the quartic del Pezzo surface Pirio, Luc Algebraic Geometry 53A60, 39B32, 14J26 (33E20, 14J45, 14L30) 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 $\big( \boldsymbol{\mathcal W}_{{\rm dP}_4}, {\bf HLog}_4\big)$. |
| title | On the 10-web by conics on the quartic del Pezzo surface |
| topic | Algebraic Geometry 53A60, 39B32, 14J26 (33E20, 14J45, 14L30) |
| url | https://arxiv.org/abs/2401.06711 |