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The Mordell Weil Groups of Cubic Pencils

Jia-Li Mo

Abstract

In this paper we study the influences of the base points of cubic pencils on the Mordell-Weil groups. Specifically, we investigate and classify the cubic pencils with 8, 7 and 6 base points in general position, and give some applications.

The Mordell Weil Groups of Cubic Pencils

Abstract

In this paper we study the influences of the base points of cubic pencils on the Mordell-Weil groups. Specifically, we investigate and classify the cubic pencils with 8, 7 and 6 base points in general position, and give some applications.
Paper Structure (6 sections, 9 theorems, 7 equations, 3 figures)

This paper contains 6 sections, 9 theorems, 7 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Proof of Theorem 1.1
  4. Proof of Theorem 1.2
  5. Proof of Theorem 1.3
  6. Applications

Key Result

theorem 1

Given $n~~(=8, 7, 6)$ points in general position in $\mathbb{P}^2$. Then in the following cases: $(1)$ 8 points are simple base points of a cubic pencil (1.1) (in fact there are 9 simple base points), $(2)$ 7 points are only simple base points of a cubic pencil (1.1), and every element of the above

Figures (3)

  • Figure 1: Dynkin diagrams of type $E_r$
  • Figure 2: The blow-up of base points
  • Figure 3: The process of contracting $(-1)$-curves

Theorems & Definitions (22)

  • theorem 1
  • remark 1
  • example 1
  • theorem 2
  • remark 2
  • theorem 3
  • theorem 4
  • definition 1
  • definition 2
  • definition 3
  • ...and 12 more