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Abundance theorem for threefolds in mixed characteristic

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Abstract

We prove the abundance theorem for arithmetic klt threefold pairs whose closed point have residue characteristic greater than 5. As a consequence, we give a sufficient condition for the asymptotic invariance of plurigenera for certain families of singular surface pairs to hold in mixed characteristic.

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References

  1. André, Y.: La conjecture du facteur direct. Publ. Math. Inst. Hautes Études Sci. 127, 71–93 (2018)

    Article  MathSciNet  Google Scholar 

  2. Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)

    Article  MathSciNet  Google Scholar 

  3. Bhatt, B.: Cohen–Macaulayness of absolute integral closures (2020). arXiv e-prints. arXiv:2008.08070

  4. Bhatt, B., Ma, L., Patakfalvi, Z., Schwede, K., Tucker, K., Waldron, J., Witaszek, J.: Globally+-regular varieties and the minimal model program for threefolds in mixed characteristic (2020). arXiv e-prints. arXiv:2012.15801

  5. Brivio, I.: Invariance of plurigenera fails in positive and mixed characteristic (2020). arXiv e-prints. arXiv:2011.10226

  6. Bernasconi, F., Tanaka, H.: On del Pezzo fibrations in positive characteristic. J. Inst. Math. Jussieu 21(1), 197–239 (2022)

    Article  MathSciNet  Google Scholar 

  7. Cossart, V., Jannsen, U., Saito, S.: Desingularization: Invariants and Strategy-Application to Dimension 2, vol. 2270. Lecture Notes in Mathematics, Springer, Cham (2020)

    Book  Google Scholar 

  8. Cossart, V., Piltant, O.: Resolution of singularities of arithmetical threefolds. J. Algebra 529, 268–535 (2019)

    Article  MathSciNet  Google Scholar 

  9. Cascini, P., Tanaka, H.: Relative semi-ampleness in positive characteristic. Proc. Lond. Math. Soc. 121(3), 617–655 (2020)

    Article  MathSciNet  Google Scholar 

  10. Cynk, S., van Straten, D.: Small resolutions and non-liftable Calabi–Yau threefolds. Manuscr. Math. 130(2), 233–249 (2009)

    Article  MathSciNet  Google Scholar 

  11. Das, O., Waldron, J.: On the abundance problem for 3-folds in characteristic \(p > 5\). Math. Z. 292(3–4), 937–946 (2019)

    Article  MathSciNet  Google Scholar 

  12. Egbert, A., Hacon, C.D.: Invariance of certain plurigenera for surfaces in mixed characteristics. Nagoya Math. J. 243, 1–10 (2021)

    Article  MathSciNet  Google Scholar 

  13. Ein, L., Lazarsfeld, R., Musta, M., Nakamaye, M., Popa, M.: Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble) 56(6), 1701–1734 (2006)

    Article  MathSciNet  Google Scholar 

  14. Floris, E., Lazić, V.: On the B-semiampleness conjecture, Épijournal Géom. Algébrique 3, Art. 12, 26 (2019)

  15. Fujino, O., Mori, S.: A canonical bundle formula. J. Differ. Geom. 56(1), 167–188 (2000)

    Article  MathSciNet  Google Scholar 

  16. Fujino, O., Tanaka, H.: On log surfaces. Proc. Jpn. Acad. Ser. A Math. Sci. 88(8), 109–114 (2012)

    Article  MathSciNet  Google Scholar 

  17. Hartshorne, R.: Algebraic Geometry, vol. 52. Graduate Texts in Mathematics. Springer, New York (1977)

    Book  Google Scholar 

  18. Hacon, C.D., McKernan, J., Xu, C.: On the birational automorphisms of varieties of general type. Ann. Math. (2) 177(3), 1077–1111 (2013)

    Article  MathSciNet  Google Scholar 

  19. Hacon, C.D., McKernan, J., Xu, C.: Boundedness of moduli of varieties of general type. J. Eur. Math. Soc. (JEMS) 20(4), 865–901 (2018)

    Article  MathSciNet  Google Scholar 

  20. Kawamata, Y.: On the extension problem of pluricanonical forms. Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), pp. 193–207 (1999)

  21. Keel, S.: Basepoint freeness for nef and big line bundles in positive characteristic. Ann. Math. (2) 149(1), 253–286 (1999)

    Article  MathSciNet  Google Scholar 

  22. Kollár, J.: Singularities of the Minimal Model Program. Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge. With a collaboration of Sándor Kovács (2013)

  23. Kollár, J.: Families of stable 3-folds in positive characteristic (2022). arXiv e-prints. arXiv:2206.02674

  24. Kollár, J. (ed.): Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992)

  25. Kollár, J.: Rational Curves on Algebraic Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 3. A Series of Modern Surveys in Mathematics. Springer, Berlin (1996)

    Google Scholar 

  26. Katsura, T., Ueno, K.: On elliptic surfaces in characteristic p. Math. Ann. 272(3), 291–330 (1985)

    Article  MathSciNet  Google Scholar 

  27. Lang, W.E.: On Enriques surfaces in characteristic p. I. Math. Ann. 265(1), 45–65 (1983)

    Article  MathSciNet  Google Scholar 

  28. Lazarsfeld, R.: Positivity in Algebraic Geometry, I. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 48. A Series of Modern Surveys in Mathematics. Springer, Berlin (2004)

    Google Scholar 

  29. Miyaoka, Y.: On the Kodaira dimension of minimal threefolds. Math. Ann. 281(2), 325–332 (1988)

    Article  MathSciNet  Google Scholar 

  30. Ma, L., Schwede, K.: Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras. Duke Math. J. 170(13), 2815–2890 (2021)

    Article  MathSciNet  Google Scholar 

  31. Shafarevich, I.R.: Lectures on Minimal Models and Birational Transformations of Two Dimensional Schemes. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 37. Tata Institute of Fundamental Research, Bombay (1966)

    Google Scholar 

  32. Shokurov, V.V.: Three-dimensional log perestroikas. Izv. Ross. Akad. Nauk Ser. Mat. 56(1), 105–203 (1992)

    MathSciNet  Google Scholar 

  33. T. Stacks Project Authors, Stacks Project

  34. Suh, J.: Plurigenera of general type surfaces in mixed characteristic. Compos. Math. 144(5), 1214–1226 (2008)

    Article  MathSciNet  Google Scholar 

  35. Tanaka, H.: Minimal models and abundance for positive characteristic log surfaces. Nagoya Math. J. 216, 1–70 (2014)

    Article  MathSciNet  Google Scholar 

  36. Tanaka, H.: Minimal model program for excellent surfaces. Ann. Inst. Fourier (Grenoble) 68(1), 345–376 (2018)

    Article  MathSciNet  Google Scholar 

  37. Tanaka, H.: Abundance theorem for surfaces over imperfect fields. Math. Z. 295(1–2), 595–622 (2020)

    Article  MathSciNet  Google Scholar 

  38. Takamatsu, T., Yoshikawa, S.: Minimal model program for semi-stable threefolds in mixed characteristic(2021). arXiv e-prints. arXiv:2012.07324

  39. Witaszek, J.: On the canonical bundle formula and log abundance in positive characteristic. Math. Ann. 381(3–4), 1309–1344 (2021)

    Article  MathSciNet  Google Scholar 

  40. Witaszek, J.: Keel’s base point free theorem and quotients in mixed characteristic. Ann. Math. (2) 195(2), 655–705 (2022)

    Article  MathSciNet  Google Scholar 

  41. Xie, L., Xue, Q.: On the existence of flips for threefolds in mixed characteristic \((0, 5)\) (2022). arXiv e-prints. arXiv:2201.08208

  42. Xu, C., Zhang, L.: Nonvanishing for 3-folds in characteristic \(p > 5\). Duke Math. J. 168(7), 1269–1301 (2019)

    Article  MathSciNet  Google Scholar 

  43. Zhang, L.: Abundance for 3-folds with non-trivial Albanese maps in positive characteristic. J. Eur. Math. Soc. (JEMS) 22(9), 2777–2820 (2020)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank F. Bongiorno, P. Cascini, J.A. Chen, E. Floris, S. Filipazzi and C.D. Hacon for useful discussions on the content of this article. We are grateful to the referee for comments and suggestions, which considerably improved the presentation of the article. FB was partially supported by the NSF under grant number DMS-1801851, by a grant from the Simons Foundation; Award Number: 256202 and by the grant \(\#\)200021/169639 from the Swiss National Science Foundation; IB was supported by the National Center for Theoretical Sciences and a grant from the Ministry of Science and Technology, grant number MOST-110-2123-M-002-005; LS is grateful to the EPSRC for his funding.

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Authors and Affiliations

  1. Chair of Algebraic Geometry (Bâtiment MA), École Polytechnique Fédérale de Lausanne, Station 8, 1015, Lausanne, Switzerland

    Fabio Bernasconi

  2. National Center for Theoretical Sciences, Taipei, 106, Taiwan

    Iacopo Brivio

  3. Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK

    Liam Stigant

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  1. Fabio Bernasconi
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  2. Iacopo Brivio
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  3. Liam Stigant
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Correspondence to Iacopo Brivio.

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Bernasconi, F., Brivio, I. & Stigant, L. Abundance theorem for threefolds in mixed characteristic. Math. Ann. 388, 141–166 (2024). https://doi.org/10.1007/s00208-022-02514-5

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