Home Lifting globally 𝐹-split surfaces to characteristic zero
Article
Licensed
Unlicensed Requires Authentication

Lifting globally 𝐹-split surfaces to characteristic zero

  • Fabio Bernasconi ORCID logo EMAIL logo , Iacopo Brivio , Tatsuro Kawakami and Jakub Witaszek
Published/Copyright: August 14, 2024

Abstract

We prove that every globally 𝐹-split surface admits an equisingular lifting over the ring of Witt vectors.

Award Identifier / Grant number: DMS-1801851

Award Identifier / Grant number: DMS-2101897

Funding source: Simons Foundation

Award Identifier / Grant number: 256202

Award Identifier / Grant number: 200021/169639

Award Identifier / Grant number: PZ00P2-21610

Award Identifier / Grant number: MOST-110-2123-M-002-005

Award Identifier / Grant number: MOST-110-2123-M-002-005

Award Identifier / Grant number: JP19J21085

Award Identifier / Grant number: JP22KJ1771

Award Identifier / Grant number: JP24K16897

Funding statement: F. Bernasconi was partially supported by the NSF under grant number DMS-1801851 and by a grant from the Simons Foundation, award number 256202, and partially by the grants #200021/169639 and PZ00P2-21610 from the Swiss National Science Foundation, I. Brivio was supported from NCTS and the grant MOST-110-2123-M-002-005. T. Kawakami was supported by JSPS KAKENHI grant numbers JP19J21085, JP22KJ1771, and JP24K16897. J. Witaszek was supported by the NSF research grant DMS-2101897.

Acknowledgements

The authors thank A. Petracci, F. Carocci, P. Cascini, C. D. Hacon, G. Martin, L. Stigant, R. Svaldi, S. Yoshikawa, T. Takamatsu, M. Nagaoka, and M. Zdanowicz for useful discussions and comments on the content of this article. The authors are also grateful to the referee for reading the manuscript very carefully and providing many valuable comments that improved our paper.

References

[1] P. Achinger, J. Witaszek and M. Zdanowicz, Global Frobenius liftability I, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 8, 2601–2648. 10.4171/jems/1063Search in Google Scholar

[2] P. Achinger, J. Witaszek and M. Zdanowicz, Global Frobenius liftability II: Surfaces and Fano threefolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 24 (2023), no. 1, 329–366. 10.2422/2036-2145.202005_003Search in Google Scholar

[3] P. Achinger and M. Zdanowicz, Some elementary examples of non-liftable varieties, Proc. Amer. Math. Soc. 145 (2017), no. 11, 4717–4729. 10.1090/proc/13622Search in Google Scholar

[4] P. Achinger and M. Zdanowicz, Serre–Tate theory for Calabi–Yau varieties, J. reine angew. Math. 780 (2021), 139–196. 10.1515/crelle-2021-0041Search in Google Scholar

[5] E. Arvidsson, F. Bernasconi and J. Lacini, On the Kawamata–Viehweg vanishing theorem for log del Pezzo surfaces in positive characteristic, Compos. Math. 158 (2022), no. 4, 750–763. 10.1112/S0010437X22007394Search in Google Scholar

[6] A. Beauville, Complex algebraic surfaces, 2nd ed., London Math. Soc. Stud. Texts 34, Cambridge University, Cambridge 1996. 10.1017/CBO9780511623936Search in Google Scholar

[7] G. Belousov, The maximal number of singular points on log del Pezzo surfaces, J. Math. Sci. Univ. Tokyo 16 (2009), no. 2, 231–238. Search in Google Scholar

[8] F. Bernasconi, Kawamata–Viehweg vanishing fails for log del Pezzo surfaces in characteristic 3, J. Pure Appl. Algebra 225 (2021), no. 11, Paper No. 106727. 10.1016/j.jpaa.2021.106727Search in Google Scholar

[9] F. Bernasconi and H. Tanaka, On del Pezzo fibrations in positive characteristic, J. Inst. Math. Jussieu 21 (2022), no. 1, 197–239. 10.1017/S1474748020000067Search in Google Scholar

[10] P. Berthelot, A. Grothendieck and L. Illusie Séminaire de géométrie algébrique du Bois Marie 1966/67, SGA 6. Théorie des intersections et théorème de Riemann–Roch, Lecture Notes in Math. 225, Springer, Berlin 1971. 10.1007/BFb0066283Search in Google Scholar

[11] R. Blache, The structure of l.c. surfaces of Kodaira dimension zero. I, J. Algebraic Geom. 4 (1995), no. 1, 137–179. Search in Google Scholar

[12] E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. 𝑝. III, Invent. Math. 35 (1976), 197–232. 10.1007/BF01390138Search in Google Scholar

[13] E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. 𝑝. II, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo (1977), 23–42. 10.1017/CBO9780511569197.004Search in Google Scholar

[14] P. Cascini and H. Tanaka, Purely log terminal threefolds with non-normal centres in characteristic two, Amer. J. Math. 141 (2019), no. 4, 941–979. 10.1353/ajm.2019.0025Search in Google Scholar

[15] P. Cascini, H. Tanaka and J. Witaszek, On log del Pezzo surfaces in large characteristic, Compos. Math. 153 (2017), no. 4, 820–850. 10.1112/S0010437X16008265Search in Google Scholar

[16] B. Conrad, O. Gabber and G. Prasad, Pseudo-reductive groups, 2nd ed., New Math. Monogr. 26, Cambridge University, Cambridge 2015. 10.1017/CBO9781316092439Search in Google Scholar

[17] R. M. Crew, Etale 𝑝-covers in characteristic 𝑝, Compos. Math. 52 (1984), no. 1, 31–45. Search in Google Scholar

[18] S. Cynk and D. van Straten, Small resolutions and non-liftable Calabi–Yau threefolds, Manuscripta Math. 130 (2009), no. 2, 233–249. 10.1007/s00229-009-0293-0Search in Google Scholar

[19] P. Deligne, Relèvement des surfaces K 3 en caractéristique nulle, Algebraic surfaces (Orsay 1976–78), Lecture Notes in Math. 868, Springer, Berlin (1981), 58–79. 10.1007/BFb0090646Search in Google Scholar

[20] P. Deligne and L. Illusie, Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247–270. 10.1007/BF01389078Search in Google Scholar

[21] J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), no. 1, 53–94. 10.1007/BF01850655Search in Google Scholar

[22] S. Ejiri, When is the Albanese morphism an algebraic fiber space in positive characteristic?, Manuscripta Math. 160 (2019), no. 1–2, 239–264. 10.1007/s00229-018-1056-6Search in Google Scholar

[23] H. Esnault and E. Viehweg, Lectures on vanishing theorems, DMV Seminar 20, Birkhäuser, Basel 1992. 10.1007/978-3-0348-8600-0Search in Google Scholar

[24] B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure and A. Vistoli, Fundamental algebraic geometry, Math. Surveys Monogr. 123, American Mathematical Society, Providence 2005. 10.1090/surv/123Search in Google Scholar

[25] O. Fujino, Minimal model theory for log surfaces, Publ. Res. Inst. Math. Sci. 48 (2012), no. 2, 339–371. 10.2977/prims/71Search in Google Scholar

[26] Y. Gongyo and S. Takagi, Surfaces of globally 𝐹-regular and 𝐹-split type, Math. Ann. 364 (2016), no. 3–4, 841–855. 10.1007/s00208-015-1238-4Search in Google Scholar

[27] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Publ. Math. Inst. Hautes Études Sci. 32 (1967), 5–361. 10.1007/BF02732123Search in Google Scholar

[28] A. Grothendieck, Séminaire de géométrie algébrique du Bois Marie 1960-61. Revêtements étales et groupe fondamental (SGA 1), Doc. Math. (Paris) 3, Société Mathématique de France, Paris 2003. Search in Google Scholar

[29] N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), no. 5, 981–996. 10.1353/ajm.1998.0037Search in Google Scholar

[30] N. Hara and K.-I. Watanabe, F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11 (2002), no. 2, 363–392. 10.1090/S1056-3911-01-00306-XSearch in Google Scholar

[31] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. 10.1007/978-1-4757-3849-0Search in Google Scholar

[32] R. Hartshorne, Deformation theory, Grad. Texts in Math. 257, Springer, New York 2010. 10.1007/978-1-4419-1596-2Search in Google Scholar

[33] L. Illusie, Complexe de de Rham–Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), no. 4, 501–661. 10.24033/asens.1374Search in Google Scholar

[34] F. Kato, Log smooth deformation theory, Tohoku Math. J. (2) 48 (1996), no. 3, 317–354. 10.2748/tmj/1178225336Search in Google Scholar

[35] N. Katz, Serre–Tate local moduli, Algebraic surfaces (Orsay 1976–7), Lecture Notes in Math. 868, Springer, Berlin (1981), 138–202. 10.1007/BFb0090648Search in Google Scholar

[36] T. Kawakami, Bogomolov–Sommese vanishing and liftability for surface pairs in positive characteristic, Adv. Math. 409 (2022), Paper No. 108640. 10.1016/j.aim.2022.108640Search in Google Scholar

[37] T. Kawakami and M. Nagaoka, Pathologies and liftability of Du Val del Pezzo surfaces in positive characteristic, Math. Z. 301 (2022), no. 3, 2975–3017. 10.1007/s00209-022-02998-6Search in Google Scholar

[38] S. Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) 149 (1999), no. 1, 253–286. 10.2307/121025Search in Google Scholar

[39] S. Keel and J. McKernan, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), no. 669, 8–153. 10.1090/memo/0669Search in Google Scholar

[40] S. L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344. 10.2307/1970447Search in Google Scholar

[41] J. Kollár, Singularities of the minimal model program, Cambridge Tracts in Math. 200, Cambridge University, Cambridge 2013. 10.1017/CBO9781139547895Search in Google Scholar

[42] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge University, Cambridge 1998. 10.1017/CBO9780511662560Search in Google Scholar

[43] J. Lacini, On rank one log del Pezzo surfaces in characteristic different from two and three, Adv. Math. 442 (2024), Paper No. 109568. 10.1016/j.aim.2024.109568Search in Google Scholar

[44] R. Laface and S. Tirabassi, On ordinary Enriques surfaces in positive characteristic, Nagoya Math. J. 245 (2022), 192–205. 10.1017/nmj.2020.36Search in Google Scholar

[45] M. Lieblich and D. Maulik, A note on the cone conjecture for K3 surfaces in positive characteristic, Math. Res. Lett. 25 (2018), no. 6, 1879–1891. 10.4310/MRL.2018.v25.n6.a9Search in Google Scholar

[46] C. Liedtke and M. Satriano, On the birational nature of lifting, Adv. Math. 254 (2014), 118–137. 10.1016/j.aim.2013.10.030Search in Google Scholar

[47] J. Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), no. 1, 151–207. 10.2307/1971141Search in Google Scholar

[48] J. Liu and L. Xie, Number of singular points on projective surfaces, preprint (2021), https://arxiv.org/abs/2103.04522. Search in Google Scholar

[49] Q. Liu, Algebraic geometry and arithmetic curves, Oxf. Grad. Texts Math. 6, Oxford University, Oxford 2002. 10.1093/oso/9780198502845.001.0001Search in Google Scholar

[50] G. Martin, Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces, Algebra Number Theory 16 (2022), no. 7, 1655–1704. 10.2140/ant.2022.16.1655Search in Google Scholar

[51] D. Maulik and B. Poonen, Néron–Severi groups under specialization, Duke Math. J. 161 (2012), no. 11, 2167–2206. 10.1215/00127094-1699490Search in Google Scholar

[52] V. B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2) 122 (1985), no. 1, 27–40. 10.2307/1971368Search in Google Scholar

[53] V. B. Mehta and V. Srinivas, Varieties in positive characteristic with trivial tangent bundle, Compos. Math. 64 (1987), no. 2, 191–212. Search in Google Scholar

[54] M. Nagaoka, Non-log liftable log del Pezzo surfaces of rank one in characteristic five, preprint (2021), https://arxiv.org/abs/2109.10558. Search in Google Scholar

[55] N. O. Nygaard, A 𝑝-adic proof of the nonexistence of vector fields on K 3 surfaces, Ann. of Math. (2) 110 (1979), no. 3, 515–528. 10.2307/1971236Search in Google Scholar

[56] N. O. Nygaard, The Tate conjecture for ordinary K 3 surfaces over finite fields, Invent. Math. 74 (1983), no. 2, 213–237. 10.1007/BF01394314Search in Google Scholar

[57] Zs. Patakfalvi and M. Zdanowicz, On the Beauville–Bogomolov decomposition in characteristic p 0 , preprint (2020), https://arxiv.org/abs/1912.12742. Search in Google Scholar

[58] Zs. Patakfalvi and M. Zdanowicz, Ordinary varieties with trivial canonical bundle are not uniruled, Math. Ann. 380 (2021), no. 3–4, 1767–1799. 10.1007/s00208-021-02165-ySearch in Google Scholar PubMed PubMed Central

[59] A. N. Rudakov and I. R. Šafarevič, Inseparable morphisms of algebraic surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 6, 1269–1307, 1439. 10.1070/IM1976v010n06ABEH001833Search in Google Scholar

[60] K. Schwede and K. E. Smith, Globally 𝐹-regular and log Fano varieties, Adv. Math. 224 (2010), no. 3, 863–894. 10.1016/j.aim.2009.12.020Search in Google Scholar

[61] J.-P. Serre, Exemples de variétés projectives en caractéristique 𝑝 non relevables en caractéristique zéro, Proc. Natl. Acad. Sci. USA 47 (1961), 108–109. 10.1073/pnas.47.1.108Search in Google Scholar PubMed PubMed Central

[62] I. Shimada, Rational double points on supersingular K 3 surfaces, Math. Comp. 73 (2004), no. 248, 1989–2017. 10.1090/S0025-5718-04-01641-2Search in Google Scholar

[63] T. K. Srivastava, On derived equivalences of K3 surfaces in positive characteristic, Doc. Math. 24 (2019), 1135–1177. 10.4171/dm/701Search in Google Scholar

[64] H. Tanaka, Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J. 216 (2014), 1–70. 10.1017/S0027763000022431Search in Google Scholar

[65] H. Tanaka, Minimal model program for excellent surfaces, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 1, 345–376. 10.5802/aif.3163Search in Google Scholar

[66] J. Witaszek, Relative semiampleness in mixed characteristic, Duke Math. J. 173 (2024), no. 9, 1631–1675. 10.1215/00127094-2023-0053Search in Google Scholar

[67] F. Yobuko, Quasi-Frobenius splitting and lifting of Calabi–Yau varieties in characteristic 𝑝, Math. Z. 292 (2019), no. 1–2, 307–316. 10.1007/s00209-018-2198-7Search in Google Scholar

[68] M. Zdanowicz, Liftability of singularities and their Frobenius morphism modulo p 2 , Int. Math. Res. Not. IMRN 2018 (2018), no. 14, 4513–4577. 10.1093/imrn/rnw297Search in Google Scholar

[69] The Stacks project authors, Stacks project, http://stacks.math.columbia.edu. Search in Google Scholar

Received: 2022-11-19
Revised: 2024-06-10
Published Online: 2024-08-14
Published in Print: 2024-11-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 16.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2024-0058/html
Scroll to top button