Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the classification of degree 1 elliptic threefolds with constant $j$-invariant

Remke Kloosterman

Abstract

We describe the possible Mordell-Weil groups for degree 1 elliptic threefold with rational base and constant $j$-invariant. Moreover, we classify all such elliptic threefolds if the $j$-invariant is nonzero. We can use this classification to describe a class of singular hypersurfaces in $\Ps(2,3,1,1,1)$ that admit no variation of Hodge structure.

On the classification of degree 1 elliptic threefolds with constant $j$-invariant

Abstract

We describe the possible Mordell-Weil groups for degree 1 elliptic threefold with rational base and constant -invariant. Moreover, we classify all such elliptic threefolds if the -invariant is nonzero. We can use this classification to describe a class of singular hypersurfaces in that admit no variation of Hodge structure.

Paper Structure

This paper contains 15 sections, 51 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Possible Mordell-Weil groups & Specialization
  4. Singularities of quartic and sextic plane curves
  5. Quartic curves
  6. Sextic curves
  7. Calculating $H^4_p(Y,{\mathbf{C}})$
  8. Dimca's method
  9. Method of Brieskorn
  10. Method of Hulek-Kloosterman
  11. Determining the Mordell-Weil rank
  12. Classification I: $j(\pi)=1728$
  13. Partial classification: case $j(\pi)=0$ and $C$ is non-reduced
  14. Case $j(\pi)=0$ and $C$ is a cuspidal curve
  15. Possible Mordell-Weil groups

Key Result

Theorem 1.1

Suppose $Y \subset {\mathbf{P}}(2,3,1,1,1)$ is a degree $6$ hypersurface, corresponding to an elliptic threefold $\pi:X \to B$, not obtained by the cone construction and not birational to a product $E\times B$. Then $\mathop{\mathrm{MW}}\nolimits(\pi)$ is one of the following

Theorems & Definitions (102)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 92 more