Mathematische Zeitschrift

, Volume 287, Issue 3–4, pp 1343–1353 | Cite as

On the base point free theorem and Mori dream spaces for log canonical threefolds over the algebraic closure of a finite field



The authors and D. Martinelli proved in (Algebra Number Theory 9(3):725–747, 2015) the base point free theorem for big line bundles on a three-dimensional log canonical projective pair defined over the algebraic closure of a finite field. In this paper, we drop the bigness condition when the characteristic is larger than five. Additionally, we discuss Mori dream spaces defined over the algebraic closure of a finite field.


Base point free theorem Semiample line bundles Positive characteristic Finite fields 

Mathematics Subject Classification

Primary 14E30 Secondary 14C20 


  1. 1.
    Artin, M.: Some numerical criteria for contractability of curves on algebraic surfaces. Am. J. Math. 84, 485–496 (1962)CrossRefMATHGoogle Scholar
  2. 2.
    Birkar, C.: Existence of flips and minimal models for 3-folds in char p. Ann. Sci. Éc. Norm. Supér. (4) 49(1), 169–212 (2016)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    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)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Birkar, C., Waldron, J.: Existence of Mori fibre spaces for 3-folds in char p (2014). arXiv:1410.4511v1
  5. 5.
    Cascini, P., Tanaka, H., Xu, C.: On base point freeness in positive characteristic. Ann. Sci. Éc. Norm. Supér. (4) 48(5), 1239–1272 (2015)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Dolgachev, I., Kondō, S.: A supersingular K3 surface in characteristic 2 and the Leech lattice. Int. Math. Res. Not. 1, 1–23 (2003)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Gongyo, Y.: On weak Fano varieties with log canonical singularities. J. Reine Angew. Math. 665, 237–252 (2012)MATHMathSciNetGoogle Scholar
  8. 8.
    Hu, Y., Keel, S.: Mori dream spaces and GIT. Mich. Math. J. 48, 331–348 (2000) Dedicated to William Fulton on the occasion of his 60th birthdayGoogle Scholar
  9. 9.
    Hacon, C.D., Xu, C.: On the three dimensional minimal model program in positive characteristic. J. Am. Math. Soc. 28(3), 711–744 (2015)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hashizume, K.: Remarks on the abundance conjecture. Proc. Jpn. Acad. Ser. A Math. Sci. 92(9), 101–106 (2016)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics. Springer, New York (1977). No. 52CrossRefGoogle Scholar
  12. 12.
    Kawamata, Y.: Flops connect minimal models. Publ. Res. Inst. Math. Sci. 44(2), 419–423 (2008)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. Adv. Stud. Pure Math. 10, 283–360 (1987)MATHMathSciNetGoogle Scholar
  14. 14.
    Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  15. 15.
    Keel, S.: Basepoint freeness for nef and big line bundles in positive characteristic. Ann. Math. (2) 149(1), 253–286 (1999)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Martinelli, D., Nakamura, Y., Witaszek, J.: On the basepoint-free theorem for log canonical threefolds over the algebraic closure of a finite field. Algebra Number Theory 9(3), 725–747 (2015)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Tanaka, H.: Minimal models and abundance for positive characteristic log surfaces. Nagoya Math. J. 216, 1–70 (2014)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Testa, D., Várilly-Alvarado, A., Velasco, M.: Big rational surfaces. Math. Ann. 351(1), 95–107 (2011)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Ueno, K.: A remark on automorphisms of Kummer surfaces in characteristic p. J. Math. Kyoto Univ. 26, 3 (1986)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Waldron, J.: The LMMP for log canonical 3-folds in char p (2016). arXiv:1603.02967v1
  21. 21.
    Xu, C.: On the base-point-free theorem of 3-folds in positive characteristic. J. Inst. Math. Jussieu 14(3), 577–588 (2015)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoMeguro-kuJapan
  2. 2.Department of MathematicsImperial College, LondonLondonUK

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