Hyperlogarithmic functional equations on del Pezzo surfaces

TL;DR

This paper establishes hyperlogarithmic functional identities on del Pezzo surfaces of degrees 1 through 6, generalizing classical identities of logarithm and dilogarithm, using Weyl group symmetries.

Contribution

It introduces a uniform method to derive hyperlogarithmic identities on del Pezzo surfaces, extending classical functional equations to higher weights.

Findings

Identifies hyperlogarithmic identities for each del Pezzo degree

Generalizes classical logarithm and dilogarithm identities

Uses Weyl group actions to establish functional equations

Abstract

For any $d\in \{1,\ldots,6\}$, we prove that the web of conics on a del Pezzo surface of degree $d$ carries a functional identity whose components are antisymmetric hyperlogarithms of weight $7-d$. Our approach is uniform with respect to $d$ and relies on classical results about the action of the Weyl group on the set of lines on the del Pezzo surface. These hyperlogarithmic functional identities are natural generalizations of the classical 3-term and (Abel's) 5-term identities satisfied by the logarithm and the dilogarithm, which correspond to the cases when $d=6$ and $d=5$ respectively.