2016/01/26 by Pietro Corvaja, Umberto Zannier, Corvaja, Pietro +1 · 6 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1601.07066
openalex publication_date 2016/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.