A structure theorem for fibrations on Delsarte surfaces

TL;DR

This paper investigates a special class of fibrations on Delsarte surfaces, showing they can be related to elliptic surfaces with three singular fibers through specific base changes, connecting to known classifications.

Contribution

It introduces Delsarte fibrations and demonstrates their relation to elliptic surfaces with three singular fibers via particular cyclic base changes.

Findings

Delsarte fibrations become pullbacks of elliptic surfaces after base change.

The base change is totally ramified at two singular points.

Every genus 1 Delsarte fibration with nonconstant j-invariant relates to Fastenberg's rational elliptic surfaces.

Abstract

In this paper we study a special class of fibrations on Delsarte surfaces. We call these fibrations Delsarte fibrations. We show that after a specific cyclic base change the fibration is the pull back of a fibration with three singular fibers, and that this second base change is completely ramified at two points where the fiber is singular. As a corollary we show that every Delsarte fibration of genus 1 with nonconstant $j$-invariant occurs as the base change of an elliptic surface from Fastenberg's list of rational elliptic surfaces with $\gamma<1$.