A structure theorem for fibrations on Delsarte 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 $\gamma<1$.