Explicit Kummer varieties of hyperelliptic Jacobian threefolds

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.