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On the Hilbert Property and the Fundamental Group of Algebraic Varieties
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.Comment: 24 page
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