Rational curves and the Hilbert property on Jacobian Kummer varieties

Gvirtz-Chen, Damián; Huang, Zhizhong

doi:10.2140/ant.2026.20.1543

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Rational curves and the Hilbert property on Jacobian Kummer varieties

Damián Gvirtz-Chen and Zhizhong Huang

Vol. 20 (2026), No. 8, 1543–1557

DOI: 10.2140/ant.2026.20.1543

Abstract

Let $K$ be a finitely generated field of characteristic zero. A conjecture by Corvaja and Zannier predicts that smooth, projective, simply connected varieties over $K$ with Zariski dense set of rational points have the Hilbert property. We confirm this conjecture for all Kummer surfaces associated to the Jacobian of a genus $2$ curve over $K$ , and in general for all Jacobian Kummer varieties associated to a hyperelliptic curve of genus $\geq 3$ of odd degree over $K$ .

Keywords

Hilbert property, thin sets, Jacobian Kummer varieties

Mathematical Subject Classification

Primary: 11G35, 14G05, 14J28

Milestones

Received: 14 May 2023

Revised: 7 February 2025

Accepted: 5 August 2025

Published: 25 September 2026

Authors

Damián Gvirtz-Chen
School of Mathematics and Statistics University of Glasgow Glasgow United Kingdom

Zhizhong Huang
Institute of Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

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Vol. 20, No. 8, 2026