A remark on a Nagell-Lutz type statement for the Jacobian of a curve of genus 2 and a (2, 3, 6) quasi-torus decomposition of a sextic with 9 cusps

TL;DR

This paper explores the limitations of a Nagell-Lutz type statement for genus 2 Jacobians over complex function fields and introduces a specific quasi-torus decomposition related to cubic dual curves.

Contribution

It demonstrates the failure of a Nagell-Lutz type theorem in a genus 2 Jacobian context and provides a novel (2, 3, 6) quasi-torus decomposition for the dual of a smooth cubic.

Findings

Nagell-Lutz type statement does not hold for certain genus 2 Jacobians over $\\mathbb{C}(t)$

Introduces a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth cubic

Highlights limitations of classical theorems in complex algebraic geometry

Abstract

In this note, an analogous statement to the Nagell-Lutz theorem does not hold for the Jacobian of a certain curve of genus 2 over $\mathbb{C}(t)$. As a by-product, we give a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth cubic.