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A structure theorem for fibrations on Delsarte surfaces

Bas Heijne, Remke Kloosterman

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 $γ<1$.

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 -invariant occurs as the base change of an elliptic surface from Fastenberg's list of rational elliptic surfaces with .

Paper Structure

This paper contains 3 sections, 13 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Delsarte surfaces
  3. Isotrivial fibrations

Key Result

Lemma 2.5

Let $S$ be a Delsarte surface. Suppose there is a nonzero vector ${\mathbf{v}}=(a,b,0,0)^T$ in ${\mathbf{Z}}^4$ such that $A{\mathbf{v}}\in \mathop{\mathrm{span}}\nolimits(1,1,1,1)^T$. Then the generic fiber of the standard fibration $\varphi:S\to {\mathbf{P}}^1$ is a rational curve.

Theorems & Definitions (39)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 29 more