On Families of Rational Elliptic Surfaces with J-Invariant Functions of Degree One

TL;DR

This paper investigates rational elliptic surfaces with degree one J-invariant functions, classifying their singular fibers, moduli space, and sections, and relates these to invariants of the regular octahedron.

Contribution

It characterizes families of rational elliptic surfaces with degree one J-invariant, describing their moduli space and sections using covering spaces and geometric invariants.

Findings

Most surfaces have four singular fibers

Moduli space is isomorphic to the projective line

Sections are rationally described by the moduli parameter

Abstract

This paper deals with a study of the rational elliptic surfaces whose $J$-invariant functions are of degree one. Almost all of these elliptic surfaces have four singular fibers, while the remaining surfaces have only three singular fibers. The moduli space of these elliptic surfaces is canonically isomorphic to the projective line by taking the $J$-values for a certain fixed type of singular fibers. Over the moduli space, we discuss our elliptic surfaces, and investigate how their sections are described by the parameter of the moduli space. By using a covering space of the moduli, we construct a family of our representative elliptic surfaces whose sections are described rationally by the parameter of the covering space. We discuss it in association with invariants of the regular octahedron.