Elliptic Fibrations and Hilbert Property

TL;DR

This paper investigates the Hilbert Property for certain algebraic varieties over number fields, providing new examples and conditions under which the property holds, including for cubic hypersurfaces, K3 surfaces, and Kummer surfaces.

Contribution

It introduces new classes of varieties with the Hilbert Property, especially smooth cubic hypersurfaces and K3 surfaces, and establishes sufficient conditions involving elliptic fibrations.

Findings

Proved the Hilbert Property for smooth cubic hypersurfaces with a rational point.

Established conditions for surfaces with multiple elliptic fibrations to have the Hilbert Property.

Demonstrated the property for certain K3 and Kummer surfaces.

Abstract

For a number field $K$, an algebraic variety $X/K$ is said to have the Hilbert Property if $X(K)$ is not thin. We are going to describe some examples of algebraic varieties, for which the Hilbert Property is a new result. The first class of examples is that of smooth cubic hypersurfaces with a $K$-rational point in $\mathbb{P}_n/K$, for $n \geq 3$. These fall in the class of unirational varieties, for which the Hilbert Property was conjectured by Colliot-Th\'el\`ene and Sansuc. We then provide a sufficient condition for which a surface endowed with multiple elliptic fibrations has the Hilbert Property. As an application, we prove the Hilbert Property of a class of K3 surfaces, and some Kummer surfaces.