Rational curves and the Hilbert Property on Jacobian Kummer varieties

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 art...

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Bibliographic Details
Published in:arXiv.org 2022-05
Main Authors: Gvirtz-Chen, Damián, Huang, Zhizhong
Format: Article
Language:English
Subjects:
Curves
Number theory
Online Access:Get full text
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Summary: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.
ISSN:2331-8422