Hyperlogarithmic functional equations on del Pezzo surfaces

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.