On the Hilbert Property and the Fundamental Group of Algebraic Varieties

TL;DR

This paper explores the Hilbert Property in algebraic varieties, linking it to fundamental group properties, providing new examples, counterexamples, and conjectures, with implications for the Inverse Galois Problem.

Contribution

It introduces new counterexamples involving Enriques surfaces and proves the Hilbert Property for a specific K3 surface, expanding understanding of the property’s relation to topology.

Findings

Counterexamples to the Hilbert Property involving Enriques surfaces

First proof of the Hilbert Property for a non-rational variety (a K3 surface)

Proposes conjectures connecting the Hilbert Property with algebraic topology

Abstract

We review, under a perspective which appears different from previous ones, the so-called Hilbert Property (HP) for an algebraic variety (over a number field); this is linked to 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, which concerns unramified covers; this link 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. We also prove the HP for a K3 surface related to the above Enriques surface, providing what appears to be the first example of a non-rational variety for which the HP can be proved. We also formulate some general conjectures relating the HP with the topology of algebraic varieties.