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

2024/01/12 by Luc Pirio, Pirio, Luc
Mathematics · Physics and Astronomy · #14J26 (33E20 #14J45 #14L30) #39B32 #53A60 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2401.06711

openalex publication_date 2024/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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≤ 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 dP₅ is a geometric model of the exceptional Bol's web and the relation \bf HLog₅ corresponds to the famous `Abel's identity' (\boldsymbol\mathcal Ab) of the dilogarithm. Bol's web together with (\boldsymbol\mathcal Ab) enjoy several remarkable properties of different kinds. We show that almost all of them admit natural generalizations to the pair ( \boldsymbol\mathcal W_\rm dP₄, \bf HLog₄).

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