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Mordell-Weil groups and Zariski triples

Jose Ignacio Cogolludo-Agustin, Remke Kloosterman

Abstract

We prove the existence of three irreducible curves $C_{12,m}$ of degree 12 with the same number of cusps and different Alexander polynomials. This exhibits a Zariski triple. Moreover we provide a set of generators for the elliptic threefold with constant $j$-invariant 0 and discriminant curve $C_{12,m}$. Finally we consider general degree $d$ base change of $C_{12d,m}$ and calculate the dimension of the equisingular deformation space.

Mordell-Weil groups and Zariski triples

Abstract

We prove the existence of three irreducible curves of degree 12 with the same number of cusps and different Alexander polynomials. This exhibits a Zariski triple. Moreover we provide a set of generators for the elliptic threefold with constant -invariant 0 and discriminant curve . Finally we consider general degree base change of and calculate the dimension of the equisingular deformation space.

Paper Structure

This paper contains 10 sections, 17 theorems, 34 equations, 1 table.

Table of Contents

  1. Construction of the curves
  2. Construction of $C_{12,0}$
  3. Construction of $C_{12,1}$
  4. Construction of $C_{12,2}$
  5. Calculation of the degree of the Alexander polynomials
  6. Alexander polynomial of $C_{12,0}$
  7. Alexander polynomial of $C_{12,1}$
  8. Alexander polynomial of $C_{12,2}$
  9. Description of Mordell-Weil groups
  10. Deformations

Key Result

Theorem 1

There exist degree 12 curves $C_{12,m}$ with $m\in \{0,1,2\}$ with precisely 32 ordinary cusps and no further singularities such that the degree of the Alexander polynomial of $C_{12,m}$ equals $2m$.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2: CogLib
  • Theorem 3: Zariski
  • Theorem 4
  • Proposition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 22 more