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Fujita’s Program and Rational Points

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Higher Dimensional Varieties and Rational Points

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 12))

Abstract

A classical theme in mathematics is the study of integral solutions of diophantine equations, that is, equations with integral coefficients. The main problems are

  • decide the existence (or nonexistence) of solutions;

  • find (some or all) solutions;

  • describe (qualitatively or quantitatively) the set of solutions.

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References

  1. V. Batyrev, The cone of effective divisors of threefolds, in: Proc. of the Intern. Conf. on Algebra (Novosibirsk, 1989), Contemp. Math., 131, Part 3, Amer. Math. Soc. (Providence, 1992), pp. 337–352.

    Google Scholar 

  2. V. Batyrev and Yu. Tschinkel, Rational points of bounded height on compactifica-tions of anisotropic tori, Intern. Math. Res. Notices, 12 (1995), 591–635.

    Article  MathSciNet  Google Scholar 

  3. V. Batyrev and Yu. Tschinkel, Rational points on some Fano cubic bundles, CR. Acad. Sci., 323 (1996), 41–46.

    MathSciNet  MATH  Google Scholar 

  4. V. Batyrev and Yu. Tschinkel. Manin’s conjecture for toric varieties, J. Algebraic Geom., 7 (1998), 15–53.

    MathSciNet  MATH  Google Scholar 

  5. V. Batyrev and Yu. Tschinkel, Tamagawa numbers of polarized algebraic varieties, in: Nombre et répartition des points de hauteur bornée (E. Peyre, ed.), Astérisque 251 (1998), 299–340.

    Google Scholar 

  6. B. Birch. Forms in many variables, Proc. Roy. Soc. Ser. A, 265 (1961–62), 245–263.

    Article  MathSciNet  Google Scholar 

  7. F. Bogomolov and Yu. Tschinkel, On the density of rational points on elliptic fibrations, J. reine angew. Math., 511 (1999), 87–93.

    MathSciNet  MATH  Google Scholar 

  8. F. Campana, Special varieties and classification theory, e-print math. AG/0110051 (2001).

    Google Scholar 

  9. A. Chambert-Loir and Yu. Tschinkel, Points of bounded height on equivariant compactifications of vector groups I, Compositio Math., 124 (2000), 65–93.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Chambert-Loir and Yu. Tschinkel, Torseurs arithmétiques et espaces fibrés, in: Rational points on algebraic varieties (E. Peyre, Yu. Tchinkel, eds.), Progress in Math., vol. 199, Birkhäuser (2001), pp. 37–69.

    Chapter  Google Scholar 

  11. A. Chambert-Loir and Yu. Tschinkel, Fonctions zêta de hauteurs des espaces fibrés, in: Rational points on algebraic varieties (E. Peyre, Yu. Tchinkel, eds.), Progress in Math., vol. 199, Birkhäuser (2001), pp. 71–116.

    Chapter  Google Scholar 

  12. A. Chambert-Loir and Yu. Tschinkel, On the distribution of rational points on equivariant compactifications of vector groups, Invent. Math., 148 (2002), 421–452.

    Article  MathSciNet  MATH  Google Scholar 

  13. J.-L. Colliot-Thélène and J.-J. Sansuc, La R-équivalence sur les tores, Ann. Sci. ENS, 10 (1977), 175–230.

    MATH  Google Scholar 

  14. J.-L. Colliot-Thélène, P. Swinnerton-Dyer and A. Skorobogatov, Double fibres and double covers: paucity of rational points. Acta Arith., 79 (1997), 113–135.

    MathSciNet  Google Scholar 

  15. C. De Concini and C. Procesi. Complete symmetric varieties, Invariant theory (Montecatini. 1982), LNM 996. Springer (1983), pp. 1–44.

    Google Scholar 

  16. R. de la Bretèche, Nombre de points de hauteur bornée sur les surfaces de del Pezzo de degré 5, to appear in Duke Math. J. (2002).

    Google Scholar 

  17. J. Denef and F. Loeser, Motivic Igusa zeta functions, J. Algebraic Geom., 7(3) (1998), 505–537.

    MathSciNet  MATH  Google Scholar 

  18. G. Fallings. Diophantine approximation on abelian varieties, Ann. of Math., 133(2) (1991), 549–576.

    Article  MathSciNet  Google Scholar 

  19. J. Franke, Yu. Manin and Yn. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math., 95 (1989), 421–435.

    Article  MathSciNet  MATH  Google Scholar 

  20. T. Fujita, On Kodaira energy of polarized varieties, J. Math. Soc. Japan, 48(1) (1996), 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Harris and Yu. Tschinkel, Rational points on quartics, Duke Math. J., 104 (2000), 477–500.

    Article  MathSciNet  MATH  Google Scholar 

  22. B. Hassett, Potential density of rational points on algebraic varieties, this volume.

    Google Scholar 

  23. B. Hassett and Yu. Tschinkel. Geometry of equivariant compactifications of (math), Intern. Math. Res. Notices, 22 (1999), 1211–1230.

    Article  MathSciNet  Google Scholar 

  24. B. Hassett and Yu. Tschinkel, On the effective cone of the moduli spaces of pointed rational curves, to appear in SISTAG Commemorative Volume: Singapore Intern. Symp. in Topology and Geometry (A. J. Berrick et al., eds.) (2001).

    Google Scholar 

  25. R. Heath-Brown, The density of rational points on cubic surfaces, Acta Arith., 272 (1997), 17–30.

    MathSciNet  Google Scholar 

  26. B. Hunt, The geometry of some special arithmetic quotients, LNM 1637, Springer (1996).

    MATH  Google Scholar 

  27. V. A. Iskovskih, Fano threefolds I, II, Izv. Akad. Nauk SSSR Ser. Mat, 41 (1977), 516–562 and

    MathSciNet  MATH  Google Scholar 

  28. V. A. Iskovskih, Fano threefolds I, II, Izv. Akad. Nauk SSSR Ser. Mat, 42 (1978), 506–549.

    MathSciNet  MATH  Google Scholar 

  29. J. Igusa. Forms of higher degree, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 59 (Bombay, 1978).

    MATH  Google Scholar 

  30. Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the Minimal Model Program, in: Algebraic Geometry, Sendai, 1985, Adv. Studies in Pure Math., vol. 10, North-Holland (1987), pp. 283–360.

    Google Scholar 

  31. J. Kollár, Low degree polynomial equations: arithmetic, geometry and topology, in: European Congress of Mathematics (Budapest, 1996), Progress in Math., vol. 168, Birkhäuser, vol. 1 (1998), pp. 255–288.

    Google Scholar 

  32. J. Kollár and Sh. Mori, Birational geometry of algebraic varieties, With the collaboration of C. H. Clemens and A. Corti. Cambridge Tracts in Mathematics 134, Cambridge University Press (1998).

    Book  MATH  Google Scholar 

  33. I. G. MacDonald, Spherical functions on a group of p-adic type, Publications of the Ramanujan Institute, No. 2. Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras (1971).

    Google Scholar 

  34. Yu. I. Manin, Cubic forms, Algebra, geometry, arithmetic, Second edition, North-Holland Mathematical Library, 4, North-Holland (Amsterdam, 1986).

    MATH  Google Scholar 

  35. Yu. Matiyasevich, Hilbert’s tenth problem: what was done and what is to be done, in: Hilberts tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Contemp. Math. vol. 270, Amer. Math. Soc. (Providence, 2000), pp. 1–47.

    Chapter  Google Scholar 

  36. Yu, Matiyasevich, Le dixième probléme de Hilbert: que peut-on faire avec les équations diophantiennes?, in: La recherche de la vérité, ACL-Ed. Kangourou (Paris, 1999), pp. 281–305.

    Google Scholar 

  37. D. McKinnon, Counting rational points on K3 surfaces, J. Number Theory, 84 (2000), 49–62.

    Article  MathSciNet  MATH  Google Scholar 

  38. Sh. Mori and Sh. Mukai, On Fano 3-folds with B 2 ≥ 2, Algebraic varieties and analytic varieties (Tokyo, 1981), 101–129, Adv. Stud. Pure Math., vol. 1, North-Holland (Amsterdam, 1983).

    Google Scholar 

  39. T. Ono. On the Tamagawa number of algebraic tori, Ann. of Math., 78(1) (1963), 47 73.

    Google Scholar 

  40. T. Ono. On the relative theory of Tamagawa numbers. Ann. of Math., 82(1) (1965). 88–111.

    Article  MathSciNet  MATH  Google Scholar 

  41. T. Ono, On Tamagawa numbers, Proc. Symp. Pure Math., vol. 9, Amer. Math. Soc. (Providence, 1966), pp. 122–132.

    Google Scholar 

  42. E. Peyre, Hauteurs et nombres de Tamagawa sur les varietés de Fano, Duke Math. J., 79 (1995), 101–218.

    Article  MathSciNet  MATH  Google Scholar 

  43. E. Peyre, Terme principal de la fonction zêta des hauteurs et torseurs universels, in: Nombre et répartition des points de hauteur bornée (E. Peyre, ed.), Astérisque. 251 (1999), 259–298.

    Google Scholar 

  44. E. Peyre and Yu. Tschinkel, Tamagawa numbers of cubic surfaces, Math. Comp., 70 (2001), 367–387.

    Article  MathSciNet  MATH  Google Scholar 

  45. J. Shalika, R. Takloo-Bighash and Yu. Tschinkel, Rational points and automorphic forms, preprint (2001).

    Google Scholar 

  46. J. Shalika, R. Takloo-Bighash and Yu. Tschinkel, Distribution of rational points on equivariant compactifications of inner forms of reductive groups, preprint (2001).

    Google Scholar 

  47. J. Shalika and Yu. Tschinkel, Height zeta functions of equivariant compactifications of the Heisenberg group, preprint (2001).

    Google Scholar 

  48. J. Shalika and Yu. Tschinkel, Height zeta functions of equivariant compactifications of unipotent groups, in preparation.

    Google Scholar 

  49. M. Strauch, Arithmetic stratifications and partial Eisenstein series, in: Rational points on algebraic varieties (E. Peyre, Yu. Tchinkel, eds.), Progress in Math., vol. 199, Birkhäuser (2001). pp. 335–355.

    Chapter  Google Scholar 

  50. M. Strauch and Yu. Tschinkel, Height zeta functions of toric bundles over flag varieties. Selecta Math., 5 (1999), 325–396.

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Tate, Fourier analysis in number fields, and Hecke’s zeta functions, in: Algebraic Number theory (J. W. S. Cassels, A. Fröhlich, eds.), Thompson (Washington, D.C., 1967). pp. 305–347.

    Google Scholar 

  52. E. B. Vinberg, The theory of convex homogeneous cones, Trans. Moscow Math. Soc., 12 (1963), 340–403.

    MATH  Google Scholar 

  53. P. Vojta, Diophantine approximations and value distribution theory, LNM 1239, Springer (1987).

    MATH  Google Scholar 

  54. R. C. Vauglian and T. D. Wooley, On a certain nonary cubic form and related equations, Duke Math. J., 80 (1995), 669–735.

    Article  MathSciNet  Google Scholar 

  55. A. Weil, Adeles and algebraic groups, Progress in Math., vol. 23, Birkhäuser (1982).

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544-1000, USA

    Yuri Tschinkel

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  1. Yuri Tschinkel
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Editors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13-15, 1053, Budapest, Hungary

    Károly Böröczky Jr.  & Tamás Szamuely  & 

  2. Department of Mathematics, Princeton University, Princeton, NJ, 08544-1000, USA

    János Kollár

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Tschinkel, Y. (2003). Fujita’s Program and Rational Points. In: Böröczky, K., Kollár, J., Szamuely, T. (eds) Higher Dimensional Varieties and Rational Points. Bolyai Society Mathematical Studies, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05123-8_9

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  • DOI: https://doi.org/10.1007/978-3-662-05123-8_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-05644-4

  • Online ISBN: 978-3-662-05123-8

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