Dessins, their delta-matroids and partial duals

TL;DR

This paper investigates the behavior of delta-matroids associated with dessins d'enfants under Galois actions, explores partial duality operations, and proves that every map has a partial dual over its field of moduli, linking topology, algebra, and tropical geometry.

Contribution

It introduces the study of delta-matroids in the context of dessins d'enfants, analyzes their transformations under Galois actions, and proves the existence of partial duals over the field of moduli.

Findings

Delta-matroids encode topological info of dessins.

Partial duals can be defined over the field of moduli.

Connections between dessins, partial duals, and tropical curves are established.

Abstract

Given a map $\mathcal M$ on a connected and closed orientable surface, the delta-matroid of $\mathcal M$ is a combinatorial object associated to $\mathcal M$ which captures some topological information of the embedding. We explore how delta-matroids associated to dessins d'enfants behave under the action of the absolute Galois group. Twists of delta-matroids are considered as well; they correspond to the recently introduced operation of partial duality of maps. Furthermore, we prove that every map has a partial dual defined over its field of moduli. A relationship between dessins, partial duals and tropical curves arising from the cartography groups of dessins is observed as well.