On subfiniteness of graded linear series

  • Huayi ChenEmail author
  • Hideaki Ikoma
Research Article


Hilbert’s fourteenth problem studies the finite generation property of the intersection of an integral algebra of finite type with a subfield of the fraction field of the algebra. It has a negative answer due to a counterexample of Nagata. We show that a subfinite version of Hilbert’s fourteenth problem has an affirmative answer. We then establish a graded analogue of this result, which permits to show that the subfiniteness of graded linear series does not depend on the function field in which we consider it. Finally, we apply the subfiniteness result to the study of geometric and arithmetic graded linear series.


Hilberts fourteenth problem Algebra of subfinite type Graded linear series Newton–Okounkov bodies 

Mathematics Subject Classification

14G40 11G30 



Huayi Chen would like to thank the Beijing International Center for Mathematical Research for their hospitality and support of his visiting position. We are grateful to Kiumars Kaveh for communications and comments. We are also grateful to anonymous referees for their suggestions and comments.


  1. 1.
    Boucksom, S., Chen, H.: Okounkov bodies of filtered linear series. Compositio Math. 147(4), 1205–1229 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bourbaki, N.: Éléments de mathématique. Algèbre. Chapitres 4 à 7. Masson, Paris (1981)Google Scholar
  3. 3.
    Chen, H.: Computing the volume function on a projective bundle over a curve. RIMS Kôkyûroku # 1745, 169–182 (2011)Google Scholar
  4. 4.
    Chen, H.: Majorations explicites des fonctions de Hilbert–Samuel géométrique et arithmétique. Math. Z. 279(1–2), 99–137 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, H.: Newton–Okounkov bodies: an approach of function field arithmetic. J. Théor. Nombres Bordeaux 30(3), 829–845 (2018) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cutkosky, S.D.: Asymptotic multiplicities of graded families of ideals and linear series. Adv. Math. 264, 55–113 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fujita, T.: On \(L\)-dimension of coherent sheaves. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(2), 215–236 (1981)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fujita, T.: Approximating Zariski decomposition of big line bundles. Kodai Math. J. 17(1), 1–3 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Les Publications Mathématiques de l’IHES, vol. 8. Paris (1961)Google Scholar
  10. 10.
    Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Les Publications Mathématiques de l’IHES, vol. 24. Paris (1965)Google Scholar
  11. 11.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)CrossRefGoogle Scholar
  12. 12.
    Kaveh, K., Khovanskii, A.: Algebraic equations and convex bodies. In: Itenberg, I., Jöricke, B., Passare, M. (eds.) Perspectives in Analysis, Geometry, and Topology. Progress in Mathematics, vol. 296, pp. 263–282. Birkhäuser/Springer, New York (2012)CrossRefGoogle Scholar
  13. 13.
    Kaveh, K., Khovanskii, A.G.: Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. 176(2), 925–978 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lazarsfeld, R.: Positivity in Algebraic Geometry. Vol. I. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 48. Springer, Berlin (2004)Google Scholar
  15. 15.
    Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. 42(5), 783–835 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8, 2nd edn. Cambridge University Press, Cambridge (1989) (Translated from the Japanese by M. Reid)Google Scholar
  17. 17.
    Mukai, S.: Counterexample to Hilbert’s fourteenth problem for three dimensional additive groups. RIMS Kôkyûroku # 1343 (2001)Google Scholar
  18. 18.
    Mukai, S.: An Introduction to Invariants and Moduli. Cambridge Studies in Advanced Mathematics, vol. 81. Cambridge University Press, Cambridge (2003) (Translated from the 1998 and 2000 Japanese editions by W.M. Oxbury)Google Scholar
  19. 19.
    Mukai, S.: Geometric realization of \(T\)-shaped root systems and counterexamples to Hilbert’s fourteenth problem. In: Popov, V.L. (ed.) Algebraic Transformation Groups and Algebraic Varieties. Encyclopaedia of Mathematical Sciences, vol. 132, pp. 123–129. Springer, Berlin (2004)CrossRefGoogle Scholar
  20. 20.
    Mumford, D.: Hilbert’s fourteenth problem—the finite generation of subrings such as rings of invariants. In: Browder, F.E. (ed.) Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics, vol. 28, pp. 431–444. American Mathematical Society, Providence (1976)Google Scholar
  21. 21.
    Nagata, M.: Addition and corrections to my paper “A treatise on the 14-th problem of Hilbert”. Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 30, 197–200 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nagata, M.: On the \(14\)-th problem of Hilbert. Amer. J. Math. 81(3), 766–772 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nagata, M.: On the fourteenth problem of Hilbert. In: Todd, A. (ed.) Proceedings of the International Congress of Mathematicians 1958, pp. 459–462. Cambridge University Press, New York (1960)Google Scholar
  24. 24.
    Okounkov, A.: Brunn–Minkowski inequality for multiplicities. Invent. Math. 125(3), 405–411 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Okounkov, A.: Why would multiplicities be log-concave? In: Duval, C., Guieu, L., Ovsienko, V. (eds.) The Orbit Method in Geometry and Physics. Progress in Mathematics, vol. 213, pp. 329–347. Birkhäuser, Boston (2003)CrossRefGoogle Scholar
  26. 26.
    Roy, D., Thunder, J.L.: Bases of number fields with small height. Rocky Mountain J. Math. 26(3), 1089–1098 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Roy, D., Thunder, J.L.: Addendum and erratum to: “An absolute Siegel’s lemma” [J. Reine Angew. Math. 476 (1996), 1–26; MR1401695 (97h:11075)]. J. Reine Angew. Math. 508, 47–51 (1999)Google Scholar
  28. 28.
    Takagi, S.: Fujita’s approximation theorem in positive characteristics. J. Math. Kyoto Univ. 47(1), 179–202 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zariski, O.: Interprétations algébrico-géométriques du quatorzième problème de Hilbert. Bull. Sci. Math. 78, 155–168 (1954)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Zariski, O.: The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface. Ann. Math. 76, 560–615 (1962)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Université Paris Diderot, Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu- Paris Rive Gauche, IMJ-PRGParisFrance
  2. 2.Faculty of EducationShitennoji UniversityHabikinoJapan

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