Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence
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Sarah Frei
Abstract
Given a smooth projective variety over a number field and an element of its Brauer group, we consider the specialization of the Brauer class at a place of good reduction for the variety and the class. We are interested in the case of K3 surfaces. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial after specialization at a set of places of positive natural density. We deduce that there exist cubic fourfolds over number fields that are conjecturally irrational, with rational reduction at a positive proportion of places. We also deduce that there are twisted derived equivalent K3 surfaces which become derived equivalent after reduction at a positive proportion of places.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1745670
Award Identifier / Grant number: DMS-1701659
Award Identifier / Grant number: DMS-1352291
Award Identifier / Grant number: DMS-1902274
Funding source: Simons Foundation
Award Identifier / Grant number: 546235
Funding statement: Sarah Frei was partially supported by NSF grant DMS-1745670. Brendan Hassett was partially supported by NSF grant DMS-1701659 and Simons Foundation Award 546235. Anthony Várilly-Alvarado was partially supported by NSF grants DMS-1352291 and DMS-1902274.
Acknowledgements
We thank Ravi Vakil for asking the third named author whether a statement like Theorem 1.5 could be true at the 2015 Arizona Winter School. We thank Nicolas Addington, Martin Bright, Jean-Louis Colliot-Thélène, Edgar Costa, Ofer Gabber, Daniel Huybrechts, Evis Ieronymou, Daniel Loughran, and Yuri Tschinkel for valuable mathematical comments and discussions, and Isabel Vogt for pointing out the reference [56]. We thank the anonymous referees for valuable comments and on earlier versions of this paper.
References
[1] N. Addington, B. Hassett, Y. Tschinkel and A. Várilly-Alvarado, Cubic fourfolds fibered in sextic del Pezzo surfaces, Amer. J. Math. 141 (2019), no. 6, 1479–1500. 10.1353/ajm.2019.0041Search in Google Scholar
[2] Y. André, On the Shafarevich and Tate conjectures for hyper-Kähler varieties, Math. Ann. 305 (1996), no. 2, 205–248. 10.1007/BF01444219Search in Google Scholar
[3]
H. Appelgate and H. Onishi,
Similarity problem over
[4] M. Artin, A. Grothendieck and J. L. Verdier, Théorie des topos et cohomologie étale des schémas. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Tome 3, Lecture Notes in Math. 305, Springer, Berlin, 1973. Search in Google Scholar
[5] J. Berg and A. Várilly-Alvarado, Odd order obstructions to the Hasse principle on general K3 surfaces, Math. Comp. 89 (2020), no. 323, 1395–1416. 10.1090/mcom/3485Search in Google Scholar
[6] F. A. Bogomolov, Sur l’algébricité des représentations l-adiques, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 15, A701–A703. Search in Google Scholar
[7] A. Cadoret and B. Moonen, Integral and adelic aspects of the Mumford–Tate conjecture, J. Inst. Math. Jussieu 19 (2020), no. 3, 869–890. 10.1017/S1474748018000233Search in Google Scholar
[8] A. Căldăraru, Nonfine moduli spaces of sheaves on K3 surfaces, Int. Math. Res. Not. 20 (2002), 1027–1056. 10.1155/S1073792802109093Search in Google Scholar
[9]
F. Charles,
The Tate conjecture for
[10] F. Charles, On the Picard number of K3 surfaces over number fields, Algebra Number Theory 8 (2014), no. 1, 1–17. 10.2140/ant.2014.8.1Search in Google Scholar
[11]
F. Charles,
Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for
[12]
E. Costa, A.-S. Elsenhans and J. Jahnel,
On the distribution of the Picard ranks of the reductions of a
[13] E. Costa and Y. Tschinkel, Variation of Néron–Severi ranks of reductions of K3 surfaces, Exp. Math. 23 (2014), no. 4, 475–481. 10.1080/10586458.2014.947054Search in Google Scholar
[14] A. J. de Jong and N. M. Katz, Monodromy and the Tate conjecture: Picard numbers and Mordell–Weil ranks in families, Israel J. Math. 120 (2000), no. A, 47–79. 10.1007/s11856-000-1271-0Search in Google Scholar
[15]
J. S. Ellenberg,
[16] A.-S. Elsenhans and J. Jahnel, Frobenius trace distributions for K3 surfaces, preprint (2021), https://arxiv.org/abs/2102.10620. Search in Google Scholar
[17] G. Faltings, p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255–299. 10.1090/S0894-0347-1988-0924705-1Search in Google Scholar
[18] S. Frei, Moduli spaces of sheaves on K3 surfaces and Galois representations, Selecta Math. (N.S.) 26 (2020), no. 1, Paper No. 6. 10.1007/s00029-019-0530-7Search in Google Scholar
[19] S. Frei, Rational reductions of geometrically irrational varieties, in preparation. Search in Google Scholar
[20] F. Grunewald and D. Segal, Some general algorithms. I. Arithmetic groups, Ann. of Math. (2) 112 (1980), no. 3, 531–583. 10.2307/1971091Search in Google Scholar
[21] D. Gvirtz and A. N. Skorobogatov, Cohomology and the Brauer groups of diagonal surfaces, preprint (2019), https://arxiv.org/abs/1905.11869 Search in Google Scholar
[22] B. Hassett, Some rational cubic fourfolds, J. Algebraic Geom. 8 (1999), no. 1, 103–114. Search in Google Scholar
[23] B. Hassett, A. Pirutka and Y. Tschinkel, Stable rationality of quadric surface bundles over surfaces, Acta Math. 220 (2018), no. 2, 341–365. 10.4310/ACTA.2018.v220.n2.a4Search in Google Scholar
[24] B. Hassett, A. Várilly-Alvarado and P. Varilly, Transcendental obstructions to weak approximation on general K3 surfaces, Adv. Math. 228 (2011), no. 3, 1377–1404. 10.1016/j.aim.2011.06.017Search in Google Scholar
[25] D. Huybrechts, Lectures on K3 surfaces, Cambridge Stud. Adv. Math. 158, Cambridge University Press, Cambridge 2016. 10.1017/CBO9781316594193Search in Google Scholar
[26] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge 2010. 10.1017/CBO9780511711985Search in Google Scholar
[27] C. Ingalls and M. Khalid, Rank 2 sheaves on K3 surfaces: a special construction, Q. J. Math. 64 (2013), no. 2, 443–470. 10.1093/qmath/has009Search in Google Scholar
[28] W. Kim and K. Madapusi Pera, 2-adic integral canonical models, Forum Math. Sigma 4 (2016), Paper No. e28. 10.1017/fms.2016.23Search in Google Scholar
[29] A. Kuznetsov, Derived categories of cubic fourfolds, Cohomological and geometric approaches to rationality problems, Progr. Math. 282, Birkhäuser, Boston (2010), 219–243. 10.1007/978-0-8176-4934-0_9Search in Google Scholar
[30] A. Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), no. 1, 251–276. 10.4007/annals.2004.159.251Search in Google Scholar
[31] M. Lieblich, D. Maulik and A. Snowden, Finiteness of K3 surfaces and the Tate conjecture, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 2, 285–308. 10.24033/asens.2215Search in Google Scholar
[32] M. Lieblich and M. Olsson, Fourier-Mukai partners of K3 surfaces in positive characteristic, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 5, 1001–1033. 10.24033/asens.2264Search in Google Scholar
[33] K. Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic, Invent. Math. 201 (2015), no. 2, 625–668. 10.1007/s00222-014-0557-5Search in Google Scholar
[34] D. W. Masser, Specializations of endomorphism rings of abelian varieties, Bull. Soc. Math. France 124 (1996), no. 3, 457–476. 10.24033/bsmf.2288Search in Google Scholar
[35] K. McKinnie, J. Sawon, S. Tanimoto and A. Várilly-Alvarado, Brauer groups on K3 surfaces and arithmetic applications, Brauer groups and obstruction problems, Progr. Math. 320, Birkhäuser/Springer, Cham (2017), 177–218. 10.1007/978-3-319-46852-5_9Search in Google Scholar
[36]
S. Mukai,
Symplectic structure of the moduli space of sheaves on an abelian or
[37]
S. Mukai,
Moduli of vector bundles on
[38]
S. Mukai,
On the moduli space of bundles on
[39] R. Noot, Abelian varieties—Galois representation and properties of ordinary reduction, Compositio Math. 97 (1995), no. 1–2, 161–171. Search in Google Scholar
[40]
D. O. Orlov,
Equivalences of derived categories and
[41] B. Poonen, Rational points on varieties, Grad. Stud. Math. 186, American Mathematical Society, Providence 2017. 10.1090/gsm/186Search in Google Scholar
[42] F. Pop, Little survey on large fields–old & new, Valuation theory in interaction, EMS Ser. Congr. Rep., European Mathematical Society, Zürich (2014), 432–463. 10.4171/149-1/21Search in Google Scholar
[43] C. Schnell, Two lectures about Mumford–Tate groups, Rend. Semin. Mat. Univ. Politec. Torino 69 (2011), no. 2, 199–216. Search in Google Scholar
[44] J.-P. Serre, Letter to K. Ribet of January 1, 1981, Œuvres. Vol. IV, Springer, Berlin (1986), 1985–1998. Search in Google Scholar
[45]
J.-P. Serre,
Un critère d’indépendance pour une famille de représentations
[46] A. N. Shankar, A. Shankar, Y. Tang and S. Tayou, Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields, preprint (2019), https://arxiv.org/abs/1909.07473v1. Search in Google Scholar
[47] T. A. Springer, Linear algebraic groups, 2nd ed., Mod. Birkhäuser Class., Birkhäuser, Boston 2009. Search in Google Scholar
[48]
S. G. Tankeev,
Surfaces of
[49]
S. G. Tankeev,
Surfaces of
[50] J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam (1968), 189–214. Search in Google Scholar
[51]
T. Terasoma,
Complete intersections with middle Picard number 1 defined over
[52] B. Totaro, Hypersurfaces that are not stably rational, J. Amer. Math. Soc. 29 (2016), no. 3, 883–891. 10.1090/jams/840Search in Google Scholar
[53] R. van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra Number Theory 1 (2007), no. 1, 1–15. 10.2140/ant.2007.1.1Search in Google Scholar
[54] A. Várilly-Alvarado, Arithmetic of del Pezzo surfaces, Birational geometry, rational curves, and arithmetic, Simons Symp., Springer, Cham (2013), 293–319. 10.1007/978-1-4614-6482-2_12Search in Google Scholar
[55]
Y. G. Zarhin,
Hodge groups of
[56] AimPL: Rational subvarieties in positive characteristic, available at http://aimpl.org/ratsubvarpos. Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Coherent categorification of quantum loop algebras: The SL(2) case
- Stable 𝔸1-connectivity over a base
- Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields
- Patchworking the Log-critical locus of planar curves
- Non-pluripolar energy and the complex Monge–Ampère operator
- Type-0 singularities in the network flow – Evolution of trees
- Categorical crystals for quantum affine algebras
- Triviality of the Hecke action on ordinary Drinfeld cuspforms of level Γ1(tn )
- Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence
Articles in the same Issue
- Frontmatter
- Coherent categorification of quantum loop algebras: The SL(2) case
- Stable 𝔸1-connectivity over a base
- Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields
- Patchworking the Log-critical locus of planar curves
- Non-pluripolar energy and the complex Monge–Ampère operator
- Type-0 singularities in the network flow – Evolution of trees
- Categorical crystals for quantum affine algebras
- Triviality of the Hecke action on ordinary Drinfeld cuspforms of level Γ1(tn )
- Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence
