On exceptional sets in Manin’s conjecture

  • Brian Lehmann
  • Sho TanimotoEmail author


In this survey paper, we study Manin’s conjecture from a geometric perspective. The focus of the paper is the recent conjectural description of the exceptional set in Manin’s conjecture due to Lehmann–Sengupta–Tanimoto. After giving an extensive background, we give a precise description of this set and compute it in many examples.


Author's contributions


The authors would like to thank Marta Pieropan and Yuri Tschinkel for a stimulating question leading to this paper and to thank Marta for many helpful comments on an earlier draft. The authors would also like to thank Brendan Hassett, Akash Sengupta, and Yuri Tschinkel for collaborations helping to shape our perspective on the a and b invariants. The first author would like to thank Jian Xiao for a useful conversation about [39]. The authors would like to thank the referee for careful reading of this paper and helpful suggestions. Brian Lehmann is supported by NSF Grant 1600875. Sho Tanimoto is partially supported by MEXT Japan, Leading Initiative for Excellent Young Researchers (LEADER).


  1. 1.
    Abdelkerim, R., Coskun, I.: Parameter spaces of Schubert varieties in hyperplane sections of Grassmannians. J. Pure Appl. Algebra 216(4), 800–810 (2012)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Andreatta, M.: Minimal model program with scaling and adjunction theory. Int. J. Math. 24(2), 1350007 (2013). 13MathSciNetzbMATHGoogle Scholar
  3. 3.
    Birkar, C., Cascini, P., Hacon, Chr D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Boucksom, S., Demailly, J.P., Paun, M., Peternell, T.: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebr. Geom. 22(2), 201–248 (2013)zbMATHGoogle Scholar
  5. 5.
    Browning, T.D., Heath-Brown, R.: Density of rational points on a quadric bundle in \({P}^3 \times {P}^3\). arXiv:1805.10715 (2018)
  6. 6.
    Birch, B.J.: Forms in many variables. Proc. R. Soc. Ser. A 265,245–263 (1961/1962)Google Scholar
  7. 7.
    Birkar, C.: Anti-pluricanonical systems on Fano varieties. arXiv:1603.05765 [math.AG] (2016)
  8. 8.
    Birkar, C.: Singularities of linear systems and boundedness of Fano varieties. arXiv:1609.05543 [math.AG] (2016)
  9. 9.
    Beheshti, R., Kumar, N.M.: Spaces of rational curves on complete intersections. Compos. Math. 149(6), 1041–1060 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Browning, T.D., Loughran, D.: Varieties with too many rational points. Math. Z. 285(3), 1249–1267 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Batyrev, V.V., Manin, YuI: Sur le nombre des points rationnels de hauteur borné des variétés algébriques. Math. Ann. 286(1–3), 27–43 (1990)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Browning, T.D.: Quantitative Arithmetic of Projective Varieties. Progress in Mathematics, vol. 277. Birkhäuser Verlag, Basel (2009)zbMATHGoogle Scholar
  13. 13.
    Browning, T.D.: Recent progress on the quantitative arithmetic of del Pezzo surfaces. In: Number Theory, Volume 6 of Series on Number Theory Applications, pp. 1–19. World Scientific Publishing, Hackensack (2010)Google Scholar
  14. 14.
    Batyrev, V.V., Tschinkel, Y.: Height zeta functions of toric varieties. J. Math. Sci. 82(1), 3220–3239 (1996). Algebraic geometry, 5MathSciNetzbMATHGoogle Scholar
  15. 15.
    Batyrev, V.V., Tschinkel, Y.: Rational points on some Fano cubic bundles. C. R. Acad. Sci. Paris Sér. I Math. 323(1), 41–46 (1996)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Batyrev, V.V., Tschinkel, Y.: Manin’s Conjecture for toric varieties. J. Algebr. Geom. 7(1), 15–53 (1998)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Batyrev, V.V., Tschinkel, Y.: Tamagawa numbers of polarized algebraic varieties. Astérisque 251, 299–340 (1998). Nombre et répartition de points de hauteur bornée (Paris, 1996)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Browning, T.D., Vishe, P.: Rational curves on smooth hypersurfaces of low degree. Algebra Number Theory 11(7), 1657–1675 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Codogni, G., Fanelli, A., Svaldi, R., Tasin, L.: Fano varieties in Mori fibre spaces. Int. Math. Res. Not. IMRN 7, 2026–2067 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Chambert-Loir, A., Tschinkel, Y.: On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math. 148(2), 421–452 (2002)MathSciNetzbMATHGoogle Scholar
  21. 21.
    de la Bretèche, R., Browning, T.D., Derenthal, U.: On Manin’s conjecture for a certain singular cubic surface. Ann. Sci. École Norm. Sup. (4) 40(1), 1–50 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    de la Bretèche, R., Browning, T.D., Peyre, E.: On Manin’s Conjecture for a family of Châtelet surfaces. Ann. Math. (2) 175(1), 297–343 (2012)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ein, L., Popa, M.: Extension of sections via adjoint ideals. Math. Ann. 352(2), 373–408 (2012)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Franke, J., Manin, YuI, Tschinkel, Y.: Rational points of bounded height on Fano varieties. Invent. Math. 95(2), 421–435 (1989)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Fujita, T.: On the structure of polarized manifolds with total deficiency one. I. J. Math. Soc. Jpn. 32(4), 709–725 (1980)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Fujita, T.: On the structure of polarized manifolds with total deficiency one. II. J. Math. Soc. Jpn. 33(3), 415–434 (1981)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Fujita, T.: Remarks on quasi-polarized varieties. Nagoya Math. J. 115, 105–123 (1989)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Fujita, T.: On Kodaira energy and adjoint reduction of polarized manifolds. Manuscr. Math. 76(1), 59–84 (1992)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Fujita, K.: Optimal bounds for the volumes of Kähler–Einstein Fano manifolds. arXiv:1508.04578 [math.AG] (2015)
  30. 30.
    Hacon, Chr D., Jiang, C.: On Fujita invariants of subvarieties of a uniruled variety. Algebr. Geom. 4(3), 304–310 (2017)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Hacon, Chr D., McKernan, J.: On Shokurov’s rational connectedness conjecture. Duke Math. J. 138(1), 119–136 (2007)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Hacon, Chr D., McKernan, J., Xu, C.: On the birational automorphisms of varieties of general type. Ann. Math. (2) 177(3), 1077–1111 (2013)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Hooley, C.: On nonary cubic forms. III. J. Reine Angew. Math. 456, 53–63 (1994)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Höring, A.: The sectional genus of quasi-polarised varieties. Arch. Math. (Basel) 95(2), 125–133 (2010)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Harris, J., Roth, M., Starr, J.: Rational curves on hypersurfaces of low degree. J. Reine Angew. Math. 571, 73–106 (2004)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Hassett, B., Tanimoto, S., Tschinkel, Y.: Balanced line bundles and equivariant compactifications of homogeneous spaces. Int. Math. Res. Not. IMRN 15, 6375–6410 (2015)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Kleiman, S.L.: Toward a numerical theory of ampleness. Ann. Math. 2(84), 293–344 (1966)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Kollár, J., Mori, Sh.: Birational Geometry of Algebraic Varieties, Volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1998). With the collaboration of Clemens, C.H., Corti, A.., Translated from the 1998 Japanese originalGoogle Scholar
  39. 39.
    Kobayashi, S., Ochiai, T.: Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ. 13, 31–47 (1973)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Kuznetsov, A.G., Prokhorov, YuG, Shramov, C.A.: Hilbert schemes of lines and conics and automorphism groups of Fano threefolds. Jpn. J. Math. 13(1), 109–185 (2018)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Le Rudulier, C.: Points algébriques de hauteur bornée sur une surface. (2014)
  42. 42.
    Lehmann, B., Sengupta, A.K., Tanimoto, S.: Geometric consistency of Manin’s Conjecture (submitted) (2018)Google Scholar
  43. 43.
    Lehmann, B., Tanimoto, S.: Geometric Manin’s Conjecture and rational curves (submitted) (2017)Google Scholar
  44. 44.
    Lehmann, B., Tanimoto, S.: On the geometry of thin exceptional sets in Manin’s Conjecture. Duke Math. J. 166(15), 2815–2869 (2017)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Lehmann, B., Tanimoto, S.: Rational curves on prime Fano threefolds of index \(1\) (submitted) (2018)Google Scholar
  46. 46.
    Lehmann, B., Tanimoto, S., Tschinkel, Y.: Balanced line bundles on Fano varieties. J. Reine Angew. Math. 743, 91–131 (2018)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Lubbes, N.: Families of bitangent planes of space curves and minimal non-fibration families. Adv. Geom. 14(4), 647–682 (2014)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Miyanishi, M., Zhang, D.-Q.: Gorenstein log del Pezzo surfaces of rank one. J. Algebra 118(1), 63–84 (1988)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Miyanishi, M., Zhang, D.-Q.: Gorenstein log del Pezzo surfaces. II. J. Algebra 156(1), 183–193 (1993)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Nakamaye, M.: Stable base loci of linear series. Math. Ann. 318(4), 837–847 (2000)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Namikawa, Y.: Smoothing Fano \(3\)-folds. J. Algebr. Geom. 6(2), 307–324 (1997)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Peyre, E.: Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke Math. J. 79(1), 101–218 (1995)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Peyre, E.: Points de hauteur bornée, topologie adélique et mesures de Tamagawa. J. Théor. Nombres Bordeaux 15(1), 319–349 (2003)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Poonen, B.: Rational Points on Varieties. Graduate Studies in Mathematics, vol. 186. American Mathematical Society, Providence (2017)zbMATHGoogle Scholar
  55. 55.
    Prokhorov, YuG: The degree of Fano threefolds with canonical Gorenstein singularities. Mat. Sb. 196(1), 81–122 (2005)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Prokhorov, YuG: The degree of \({\mathbb{Q}}\)-Fano threefolds. Mat. Sb. 198(11), 153–174 (2007)MathSciNetGoogle Scholar
  57. 57.
    Reid, M.: The complete intersection of two or more quadrics. Thesis. (1972)
  58. 58.
    Riedl, E., Yang, D.: Kontsevich spaces of rational curves on Fano hypersurfaces. arXiv:1409.3802 [math.AG] (2016), to appear in J. Reine Agnew. Math
  59. 59.
    Salberger, P.: Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Astérisque 251, 91–258 (1998). Nombre et répartition de points de hauteur bornée (Paris, 1996)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Schläfli, L.: On the distribution of surfaces of the third order into species. Philos. Trans. R. Soc. 153, 193–247 (1864)Google Scholar
  61. 61.
    Schanuel, S.H.: Heights in number fields. Bull. Soc. Math. Fr. 107(4), 433–449 (1979)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Sengupta, A.K.: Manin’s \(b\)-constant in families. arXiv:1708.05447 [math.AG] (2017)
  63. 63.
    Sengupta, A.K.: Manin’s conjecture and the Fujita invariant of finite covers. arXiv:1712.07780 (2017)
  64. 64.
    Sommese, A.J.: On the adjunction theoretic structure of projective varieties. In: Complex Analysis and Algebraic Geometry (Göttingen, 1985). Lecture Notes in Mathematics, vol. 1194, pp. 175–213. Springer, Berlin (1986)Google Scholar
  65. 65.
    Shalika, J., Tschinkel, Y.: Height zeta functions of equivariant compactifications of unipotent groups. Commun. Pure Appl. Math. 69(4), 693–733 (2016)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Shalika, J., Takloo-Bighash, R., Tschinkel, Y.: Rational points on compactifications of semi-simple groups. J. Am. Math. Soc. 20(4), 1135–1186 (2007)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Vaughan, R.C.: On Waring’s problem for cubes. J. Reine Angew. Math. 365, 122–170 (1986)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Xu, C.: Finiteness of algebraic fundamental groups. Compos. Math. 150(3), 409–414 (2014)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Ye, Q.: On Gorenstein log del Pezzo surfaces. Jpn. J. Math. (N.S.) 28(1), 87–136 (2002)MathSciNetzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsBoston CollegeChestnut HillUSA
  2. 2.Department of Mathematics, Faculty of ScienceKumamoto UniversityKumamotoJapan
  3. 3.Priority Organization for Innovation and ExcellenceKumamoto UniversityKumamotoJapan

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