Webs by conics on del Pezzo surfaces and hyperlogarithmic functional identities

TL;DR

This paper establishes a series of hyperlogarithmic functional identities associated with webs by conics on del Pezzo surfaces of degrees 2 to 6, generalizing classical logarithmic identities.

Contribution

It introduces a uniform approach to derive hyperlogarithmic identities on del Pezzo surfaces, extending classical Abel relations to higher weights and degrees.

Findings

Identifies hyperlogarithmic identities for del Pezzo surfaces of degrees 2 to 6.

Generalizes classical 3-term and 5-term identities to higher weights.

Connects geometric configurations with functional equations of hyperlogarithms.

Abstract

For $d$ ranging from 2 to 6, we prove that the web by conics naturally defined on any smooth del Pezzo surface of degree $d$ carries an interesting functional identity whose components all are a certain antisymmetric hyperlogarithm of weight $7-d$. Our approach is uniform with respect to $d$ and at the end relies on classical results about the action of Weyl groups on the set of lines contained in the considered del Pezzo surface. This series of `del Pezzo's hyperlogarithmic functional identities' is a natural generalization of the famous and well-know 3-term and 5-term identities of the logarithm and dilogarithm ('Abel's relation') which correspond to the cases when $d=6$ and $d=5$ respectively. This text ends with a section containing several questions and some possibly interesting perspectives.