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On the Hilbert property and the fundamental group of algebraic varieties

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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 \({\mathbb {C}}\)) 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.

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Notes

  1. These links appear to have never been remarked before, and usually books and expositions dealt with these concepts in separate parts.

  2. In fact, it suffices to assume that this set is always nonempty; indeed, if the set happens to be contained in a proper Zariski closed subset, say defined by \(f=0\), it suffices to consider a further cover with function field \(k(X)(f^{1/n})\) for suitable n to lift all remaining points in X(k), thus making empty the complement with respect to this further cover.

  3. Such algebraic groups are geometrically rational varieties, but possibly not over k; however, they are always k-unirational.

  4. It is easy to see that \(E^{(2)}\) is birationally isomorphic to \(E\times {\mathbb {P}}_1\).

  5. There are seven possibilities for a nontrivial such G: see the classification in [3], ch. VI.

  6. However some authors adopt the converse terminology.

  7. The variety \(Y'\) may be obtained e.g. by taking on affine pieces the integral closure of the affine ring of X in the function field k(Y), and then by gluing these affine parts.

  8. We do not consider here multiplicities; see [6] for this.

  9. It was proved by Serre [32] that different embeddings \(k\hookrightarrow {\mathbb {C}}\) can produce different fundamental groups; however the properties relevant for us are independent of this embedding, and the same holds for the profinite completion of the fundamental group.

  10. One of Catanese’s examples starts from a product \({\mathbb {P}}_1\times E\) where E is an elliptic curve, taking the quotient by the equivalence relation \((t_1,p_1)\sim (t_2,p_2)\) if and only if \(t_1=t_2=0\) and \(p_2 = - p_1\) or \((t_1,p_1)=(t_2,p_2)\). We obtain a variety X which may be shown to be simply connected. Its normalization is just the original \({\mathbb {P}}_1\times E\), whose fundamental group is \({\mathbb {Z}}^2\).

  11. On the contrary, if a normal model is not simply connected, the same holds for a desingularization, as can be seen on taking the pullback of a possible cover to the smooth model.

  12. These surfaces were first exhibited by Enriques: in 1894 he communicated in a letter to Castelnuovo a relevant example of a sextic, which showed that the vanishing of the irregularity and geometric genus were not sufficient for rationality; see [15]. Afterwards, in 1896, Castelnuovo found his celebrated rationality criterion, that the vanishing of the irregularity and the second plurigenus are indeed sufficient. We thank P. Oliverio for these references.

  13. This version is probably known, though usually not explicitly mentioned. See [11] for the case of cyclic covers and [12] for a general statement.

  14. We thank T. Yasuda for remarks leading to this formulation, more precise than a former one.

  15. There are also concepts of strong a. p.—different from WAP only for integral points - and of hyper-weak a. p., this last one being introduced by Harari, see [20].

  16. More generally, his conjecture should include every rationally connected variety with a dense set of rational points.

  17. We thank Colliot-Thélène for this (and related) reference(s).

  18. This case is also linked with other questions concerning the Brauer Manin obstruction: it is not known whether for rationally connected varieties this is the only obstruction to weak approximation: if this was the case, then HP for smooth rationally connected varieties would hold; we thank Borovoi for explaining us this link

  19. Historically, the case of curves of genus one was settled previously by Néron, [26], via a method based on heights estimates on abelian varieties, subsequently much improved by Mumford; still another proof for this case can be deduced from the results in [41].

  20. This happens in considerable generality, as shown e.g. by Serre, after Faltings’ proof of certain Tate conjectures.

  21. They may be defined as the points which do not reduce to the divisor at infinity for any prime outside S.

  22. It is immaterial that we require that the points are integral also in the covers, since one can change anyway slightly a cover to get a finite map and ensure integrality of rational points sent to integral ones.

  23. It is worth noticing that, somewhat in the converse direction, no example seems to be known of a smooth simply connected projective surface where the rational points can be proved to be not Zariski-dense over any number field; this should be expected e.g. for smooth hypersurfaces in \({\mathbb {P}}_3\) of large enough degree (which are of general type). Instead, for integral points on affine surfaces such an example may be found in the paper [13] by the authors: see Thm. 3 and Cor. 2 therein.

  24. He actually proved the density of rational points in the euclidean topology, inside the set of real points

  25. Here we could also use the ‘tangent process’, i.e. intersecting the cubic with the tangent at the previous point; or else we could intersect the cubic with the line. However this last method, though sufficient for our purposes, would produce rational points only for a ‘thin’ subset of the pencil.

  26. In fact, if we define \(\lambda \) by the same formula on the whole \({\mathbb {P}}_3\), then it is not defined at any point of L. However, its restriction to F can be continued to a regular map on the whole surface.

  27. G. Lido has found an alternative argument, almost avoiding calculations and using the intersection product of F.

  28. This double fibration by elliptic curves yields two sets \(\{[m]_i:m\in {\mathbb {Z}}\}, i=1,2\) of rational endomorphisms of F obtained by integer multiplication on the two families; on using the compositions \([m]_1\circ [n]_2\), \(m,n\in {\mathbb {Z}}\), applied to a given rational point, we again obtain a Zariski-dense set of rational points.

  29. We note that considering a single fibration (and merely the rational points coming from the above sections) would not suffice in absence of additional information. Indeed, under e.g. the first fibration, F may be seen as an elliptic curve over \({\mathbb {Q}}(\lambda )\), with a point of infinite order defined over \({\mathbb {Q}}(\lambda )\); then the group generated by this point would lift to the union of the two covers of this elliptic curve obtained by division by 2 followed by suitable translations (as in the weak Mordell–Weil).

  30. Actually, Bogomolov and Tschinkel [4] proved that after a finite extension of their field of definition, every Enriques surface has a Zariski-dense set of rational points (potential density of rational points).

  31. \(F'\) admits a unique smooth minimal model which can be obtained as follows: after blowing up the eight fixed points of \(\sigma ^2\) on F, one obtains a smooth quotient \(\hat{F'}\) of the blown-up surface \(\hat{F}\); the quotient map \(\hat{F} \rightarrow \hat{F'}\) ramifies exactly on the eight exceptional divisors, whose sum constitutes the canonical divisor of \(\hat{F}\); this is in accordance with the fact that the canonical divisor of \(\hat{F'}\) vanishes.

  32. The argument yields directly that \({\mathcal {E}}({\mathbb {Q}})\) lifts to \({\mathcal {F}}({\mathbb {Q}}(i))\), which is analogue to the standard Chevalley–Weil theorem. The present device, on replacing one cover by two covers to maintain the ground field, is similar to the above version of Chevalley–Weil.

  33. We have preferrred to use this model of a hypersurface of \({\mathbb {P}}_3\) both for simplicity and also because it illustrates some subtleties related to the fundamental group of different models.

  34. J. Demeio has carried out in detail this proof.

  35. These reductions rely on the fact that a smooth model of X is known to be simply connected, and all its—ramified—covers not factoring via the canonical map \(A\rightarrow X\) come from ramified covers of A.

  36. A priori the torsion order would be absolutely bounded, because a rational torsion point over \({\mathbb {Q}}(u_1,u_2)\) cannot have ‘too large’ order, due to the Galois theory of Fricke and Weber, no need to use the deeper arithmetical results here. But we do not even need to use this, or to perform computations: indeed, \(Z_{u_1,u_2}\) depends only on \(w:=f_2(u_2)/f_1(u_1)\), whereas we may vary both \(u_1,u_2\); then we would obtain a continuous family of points of finite order on a same elliptic curve, which is impossible.

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Acknowledgements

We thank M. Borovoi, F. Catanese, J.-L. Colliot-Thélène, P. Dèbes, A. Fehm, M. Fried, D. Harari, G. Lido, P. Oliverio, P. Sarnak and T. Yasuda for their interest and important comments. We also acknowledge the support of the ERC Grant ‘Diophantine Problems’ in this research, and the University of Konstanz for invitation to a school where the present paper started to be conceived. We further thank the referee for a careful review and several helpful remarks.

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  1. Università di Udine, Via delle Scienze, 33100, Udine, Italy

    Pietro Corvaja

  2. Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126, Pisa, Italy

    Umberto Zannier

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Corvaja, P., Zannier, U. On the Hilbert property and the fundamental group of algebraic varieties. Math. Z. 286, 579–602 (2017). https://doi.org/10.1007/s00209-016-1775-x

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