Real trigonal curves and real elliptic surfaces of type I

TL;DR

This paper investigates real trigonal curves and elliptic surfaces of type I, using real dessins d'enfants to classify maximally inflected cases and analyze their deformation and singular fiber configurations.

Contribution

It introduces a real version of dessins d'enfants to classify maximally inflected trigonal curves of type I and provides a complete classification for rational bases.

Findings

Complete classification of maximally inflected trigonal curves of type I over rational bases.

Description of real Jacobian elliptic surfaces of type I with all real singular fibers.

Development of a real dessins d'enfants framework for analyzing these curves and surfaces.

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 \emph{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.