Explicit Kummer varieties of hyperelliptic Jacobian threefolds

TL;DR

This paper explicitly constructs the Kummer variety for genus 3 hyperelliptic Jacobians over fields with characteristic not 2, providing concrete equations and duplication maps.

Contribution

It provides explicit equations and constructions for the Kummer variety and duplication map of genus 3 hyperelliptic Jacobians, advancing computational approaches.

Findings

Explicit equations for the Kummer variety

Homogeneous quartic polynomials representing duplication

Construction valid over fields with characteristic not 2

Abstract

We explicitly construct the Kummer variety associated to the Jacobian of a hyperelliptic curve of genus 3 that is defined over a field of characteristic not equal to 2 and has a Weierstra{\ss} point defined over the same field. We also construct homogeneous quartic polynomials on the Kummer variety and show that they represent the duplication map.