Rational curves and the Hilbert Property on Jacobian Kummer varieties

TL;DR

This paper proves the Hilbert Property for Jacobian Kummer varieties of genus 2 hyperelliptic curves over fields of characteristic zero, extending previous results to a broader class of algebraic varieties.

Contribution

It establishes the Hilbert Property for all Jacobian Kummer surfaces associated to genus 2 hyperelliptic curves of odd degree, generalizing prior work on Kummer surfaces.

Findings

Proves the Hilbert Property for Jacobian Kummer surfaces of genus 2.

Extends the Hilbert Property to hyperelliptic curves of genus ≥ 2 of odd degree.

Supports the conjecture relating rational points and the Hilbert Property.

Abstract

A conjecture by Corvaja and Zannier predicts that smooth, projective, simply connected varieties over a number field with Zariski dense set of rational points have the Hilbert Property; this was proved by Demeio for Kummer surfaces which are associated to products of two elliptic curves. In this article, over a finitely generated field of characteristic zero, we establish the Hilbert Property for all Kummer surfaces associated to the Jacobian of a genus $2$ curve. In general we prove that all Jacobian Kummer varieties associated to a hyperelliptic curve of genus $\geq 2$ of odd degree also have the Hilbert Property.