Real trigonal curves and real elliptic surfaces of type I

Abstract

We study real trigonal curves and elliptic surfaces of type I (over a base of an arbitrary genus) and their fiberwise equivariant deformations. The principal tool is a real version of Grothendieck's dessins d'enfants. We give a description of maximally inflected trigonal curves of type I in terms of the combinatorics of sufficiently simple graphs and, in the case of the rational base, obtain a complete classification of such curves. As a consequence, these results lead to conclusions concerning real Jacobian elliptic surfaces of type I with all singular fibers real. © De Gruyter 2014.