On the Hilbert property and the fundamental group of algebraic varieties

Abstract

In this paper we link the so-called Hilbert property (HP) for an algebraic variety (over a number field) with its fundamental group, in a perspective which appears new. (The notion of HP comes from Hilbert’s Irreducibility Theorem and has important implications, for instance towards the Inverse Galois Problem.) We shall observe that the HP is in a sense ‘opposite’ to the Chevalley–Weil Theorem. This shall immediately entail the result that the HP can possibly hold only for simply connected varieties (in the appropriate sense). In turn, this leads to new counterexamples to the HP, involving Enriques surfaces. In this view, we shall also formulate an alternative related property, possibly holding in full generality, and some conjectures which unify part of what is known in the topic. These predict that for a variety with a Zariski-dense set of rational points, the validity of the HP, which is defined arithmetically, is indeed of purely topological nature. Also, a consequence of these conjectures would be a positive solution of the Inverse Galois Problem. In the paper we shall also prove the HP for a K3 surface related to the above Enriques surface, providing what appears to be the first example of a variety non rational (over (Formula presented.)) for which the HP can be proved. In an Appendix we shall further discuss, among other things, the HP also for Kummer surfaces. All of this basically exhausts the study of the HP for surfaces with a Zariski-dense set of rational points

Elliptic Fibrations and the Hilbert Property

“…The examples are produced starting from a construction presented in [5], by Garbagnati and Salgado. Finally, we prove the Hilbert Property for some Kummer surfaces, for which the Hilbert Property was suggested to be true by Corvaja and Zannier [3].…”

“…The first to have used multiple elliptic fibrations to prove the Hilbert Property are Corvaja and Zannier, who proved it for the Fermat surface x 4 + y 4 = z 4 + w 4 [3,Theorem 1.4].…”

“…Let us denote by A * 4 the dual affine space of A 4 , minus the origin. To each element h ∈ A * 4 corresponds a hyperplane H(h) ⊂ P 3 in a canonical way 3…”

“…A normal variety that is not geometrically integral over the base field K does not have any K-rational points. 3 Namely…”

Non-rational varieties with the Hilbert Property

“…Theorem 1.4 is essentially an elaboration and generalization of the ideas of Corvaja and Zannier [CZ16]. There are, nevertheless, some key differences in the proof.…”