A Hilbert irreducibility theorem for Enriques surfaces

Gvirtz-Chen, Damián; Mezzedimi, Giacomo

doi:10.1090/tran/8831

A Hilbert irreducibility theorem for Enriques surfaces
HTML articles powered by AMS MathViewer

by Damián Gvirtz-Chen and Giacomo Mezzedimi;

Trans. Amer. Math. Soc. 376 (2023), 3867-3890

DOI: https://doi.org/10.1090/tran/8831

Published electronically: March 21, 2023

HTML | PDF | Request permission

Abstract:

We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension $0$. Bounding the rank of this object, we prove that a conjecture by Campana [Ann. Inst. Fourier (Grenoble) 54 (2004), pp. 499–630] and Corvaja–Zannier [Math. Z. 286 (2017), pp. 579–602] holds for Enriques surfaces, as well as K3 surfaces of Picard rank $\geq 6$ apart from a finite list of geometric Picard lattices.

Concretely, we prove that such surfaces over finitely generated fields of characteristic $0$ satisfy the weak Hilbert Property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded.

References

Michael Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485–496. MR 146182, DOI 10.2307/2372985
Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225, DOI 10.1007/978-3-642-57739-0
Neil Berry, Artūras Dubickas, Noam D. Elkies, Bjorn Poonen, and Chris Smyth, The conjugate dimension of algebraic numbers, Q. J. Math. 55 (2004), no. 3, 237–252. MR 2082091, DOI 10.1093/qjmath/55.3.237
F. A. Bogomolov and Yu. Tschinkel, Density of rational points on Enriques surfaces, Math. Res. Lett. 5 (1998), no. 5, 623–628. MR 1666840, DOI 10.4310/MRL.1998.v5.n5.a6
F. A. Bogomolov and Yu. Tschinkel, Density of rational points on elliptic $K3$ surfaces, Asian J. Math. 4 (2000), no. 2, 351–368. MR 1797587, DOI 10.4310/AJM.2000.v4.n2.a6
Simon Brandhorst and Giacomo Mezzedimi, Borcherds lattices and K3 surfaces of zero entropy, arXiv:2211.09600, 2022.
Simon Brandhorst, Serkan Sonel, and Davide Cesare Veniani, Idoneal genera and K3 surfaces covering an Enriques surface, arXiv:2003.08914, 2020.
Anna Cadoret and Akio Tamagawa, Uniform boundedness of $p$-primary torsion of abelian schemes, Invent. Math. 188 (2012), no. 1, 83–125. MR 2897693, DOI 10.1007/s00222-011-0343-6
Frédéric Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 499–630 (English, with English and French summaries). MR 2097416, DOI 10.5802/aif.2027
Serge Cantat, Dynamique des automorphismes des surfaces projectives complexes, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 10, 901–906 (French, with English and French summaries). MR 1689873, DOI 10.1016/S0764-4442(99)80294-8
Pietro Corvaja, Julian Lawrence Demeio, Ariyan Javanpeykar, Davide Lombardo, and Umberto Zannier, On the distribution of rational points on ramified covers of abelian varieties, Compos. Math. 158 (2022), no. 11, 2109–2155. MR 4519542, DOI 10.1112/s0010437x22007746
Pietro Corvaja and Umberto Zannier, On the Hilbert property and the fundamental group of algebraic varieties, Math. Z. 286 (2017), no. 1-2, 579–602. MR 3648511, DOI 10.1007/s00209-016-1775-x
François R. Cossec, On the Picard group of Enriques surfaces, Math. Ann. 271 (1985), no. 4, 577–600. MR 790116, DOI 10.1007/BF01456135
François R. Cossec and Igor V. Dolgachev, Enriques surfaces. I, Progress in Mathematics, vol. 76, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 986969, DOI 10.1007/978-1-4612-3696-2
Julian Lawrence Demeio, Non-rational varieties with the Hilbert property, Int. J. Number Theory 16 (2020), no. 4, 803–822. MR 4093384, DOI 10.1142/S1793042120500414
Julian Lawrence Demeio, Elliptic fibrations and the Hilbert property, Int. Math. Res. Not. IMRN 13 (2021), 10260–10277. MR 4283578, DOI 10.1093/imrn/rnz108
Igor V. Dolgachev and Shigeyuki Kondō, Enriques surfaces. ii, September 2022, book draft available at http://www.math.lsa.umich.edu/~idolga/EnriquesTwo.pdf.
Wolfgang Ebeling, Lattices and codes, 3rd ed., Advanced Lectures in Mathematics, Springer Spektrum, Wiesbaden, 2013. A course partially based on lectures by Friedrich Hirzebruch. MR 2977354, DOI 10.1007/978-3-658-00360-9
Gerhard Frey and Moshe Jarden, Approximation theory and the rank of abelian varieties over large algebraic fields, Proc. London Math. Soc. (3) 28 (1974), 112–128. MR 337997, DOI 10.1112/plms/s3-28.1.112
Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 3, Société Mathématique de France, Paris, 2003 (French). Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61]; Directed by A. Grothendieck; With two papers by M. Raynaud; Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. MR 2017446
Daniel Huybrechts, Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, vol. 158, Cambridge University Press, Cambridge, 2016. MR 3586372, DOI 10.1017/CBO9781316594193
Jong Hae Keum, Every algebraic Kummer surface is the $K3$-cover of an Enriques surface, Nagoya Math. J. 118 (1990), 99–110. MR 1060704, DOI 10.1017/S0027763000003019
J. Keum and D.-Q. Zhang, Fundamental groups of open $K3$ surfaces, Enriques surfaces and Fano 3-folds, J. Pure Appl. Algebra 170 (2002), no. 1, 67–91. MR 1896342, DOI 10.1016/S0022-4049(01)00110-4
Shigeyuki Kond\B{o}, Enriques surfaces with finite automorphism groups, Japan. J. Math. (N.S.) 12 (1986), no. 2, 191–282. MR 914299, DOI 10.4099/math1924.12.191
Shigeyuki Kond\B{o}, Algebraic $K3$ surfaces with finite automorphism groups, Nagoya Math. J. 116 (1989), 1–15. MR 1029967, DOI 10.1017/S0027763000001653
Masato Kuwata and Lan Wang, Topology of rational points on isotrivial elliptic surfaces, Internat. Math. Res. Notices 4 (1993), 113–123. MR 1214702, DOI 10.1155/S107379289300011X
Kuan-Wen Lai and Masahiro Nakahara, Uniform potential density for rational points on algebraic groups and elliptic K3 surfaces, Int. Math. Res. Not. IMRN 23 (2022), 18541–18588. MR 4519151, DOI 10.1093/imrn/rnab237
Gebhard Martin, Enriques surfaces with finite automorphism group in positive characteristic, Algebr. Geom. 6 (2019), no. 5, 592–649. MR 4009175, DOI 10.14231/ag-2019-027
Gebhard Martin, Giacomo Mezzedimi, and Davide Cesare Veniani, Enriques surfaces of non-degeneracy 3, arXiv:2203.08000, 2022.
William E. Lang, On Enriques surfaces in characteristic $p$. I, Math. Ann. 265 (1983), no. 1, 45–65. MR 719350, DOI 10.1007/BF01456935
Rick Miranda, The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica. [Doctorate in Mathematical Research], ETS Editrice, Pisa, 1989. MR 1078016
V. V. Nikulin, Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by $2$-reflections. Algebro-geometric applications, J. Sov. Math. 22 (1983), 1401–1475.
V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
Viacheslav V. Nikulin, Elliptic fibrations on $\rm K3$ surfaces, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 1, 253–267. MR 3165023, DOI 10.1017/S0013091513000953
Luis Ribes, Introduction to profinite groups, Travaux mathématiques. Vol. XXII, Trav. Math., vol. 22, Fac. Sci. Technol. Commun. Univ. Luxemb., Luxembourg, 2013, pp. 179–230. MR 3204848
Matthias Schütt, Divisibilities among nodal curves, Math. Res. Lett. 25 (2018), no. 4, 1359–1368. MR 3882167, DOI 10.4310/MRL.2018.v25.n4.a14
Matthias Schütt and Tetsuji Shioda, Elliptic surfaces, Algebraic geometry in East Asia—Seoul 2008, Adv. Stud. Pure Math., vol. 60, Math. Soc. Japan, Tokyo, 2010, pp. 51–160. MR 2732092, DOI 10.2969/aspm/06010051
Jean-Pierre Serre, Topics in Galois theory, 2nd ed., Research Notes in Mathematics, vol. 1, A K Peters, Ltd., Wellesley, MA, 2008. With notes by Henri Darmon. MR 2363329
Ali Sinan Sertöz, Which singular $K3$ surfaces cover an Enriques surface, Proc. Amer. Math. Soc. 133 (2005), no. 1, 43–50. MR 2085151, DOI 10.1090/S0002-9939-04-07666-X
Ichiro Shimada and De-Qi Zhang, Classification of extremal elliptic $K3$ surfaces and fundamental groups of open $K3$ surfaces, Nagoya Math. J. 161 (2001), 23–54. MR 1820211, DOI 10.1017/S002776300002211X
Tamás Szamuely, Galois groups and fundamental groups, Cambridge Studies in Advanced Mathematics, vol. 117, Cambridge University Press, Cambridge, 2009. MR 2548205, DOI 10.1017/CBO9780511627064
È. B. Vinberg, Classification of 2-reflective hyperbolic lattices of rank 4, Tr. Mosk. Mat. Obs. 68 (2007), 44–76 (Russian, with Russian summary); English transl., Trans. Moscow Math. Soc. (2007), 39–66. MR 2429266, DOI 10.1090/s0077-1554-07-00160-4
Xun Yu, Elliptic fibrations on K3 surfaces and Salem numbers of maximal degree, J. Math. Soc. Japan 70 (2018), no. 3, 1151–1163. MR 3830803, DOI 10.2969/jmsj/75907590
Xun Yu, K3 surface entropy and automorphism groups, arXiv:2211.07526, 2022.