Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1357–1379 | Cite as

Newton–Okounkov bodies on projective bundles over curves

  • Pedro MonteroEmail author


In this article, we study Newton–Okounkov bodies on projective vector bundles over curves. Inspired by Wolfe’s estimates used to compute the volume function on these varieties, we compute all Newton–Okounkov bodies with respect to linear flags. Moreover, we characterize semi-stable vector bundles over curves via Newton–Okounkov bodies.

Mathematics Subject Classification

14C20 14H60 14M99 



I would like to express my gratitude to my thesis supervisors, Stéphane Druel and Catriona Maclean, for their advice, helpful discussions and encouragement throughout the preparation of this article. I also thank Bruno Laurent, Laurent Manivel and Bonala Narasimha Chary for fruitful discussions. Finally, I would like to thank the anonymous referee for a very helpful and detailed report.


  1. 1.
    Bădescu, L.: In: by V. Maşek., (ed.) Algebraic surfaces. Transl. from the Romanian. Springer, New York (2001)Google Scholar
  2. 2.
    Biswas, I., Hogadi, A., Parameswaran, A.J.: Pseudo-effective cone of Grassmann bundles over a curve. Geom. Dedicata 172, 69–77 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boucksom, S.: Corps d’Okounkov. Séminaire Bourbaki 65, 1–38 (2012)Google Scholar
  4. 4.
    Biswas, I., Parameswaran, A.J.: Nef cone of flag bundles over a curve. Kyoto J. Math. 54(2), 353–366 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brion, M.: Lectures on the Geometry of Flag Varieties. Birkhäuser, Trends in Mathematics. Basel (2005)CrossRefGoogle Scholar
  6. 6.
    Butler, D.C.: Normal generation of vector bundles over a curve. J. Differ. Geom. 39(1), 1–34 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, H.: Computing volume function on projective bundle over a curve. Hodge theory and algebraic geometry, RIMS Kôkyûroku 1745, 169–182 (2011)Google Scholar
  8. 8.
    Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M., Popa, M.: Asymptotic invariants of base loci. Ann. Inst. Fourier 56(6), 1701–1734 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M., Popa, M.: Restricted volumes and base loci of linear series. Am. J. Math. 131(3), 607–651 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fulton, W.: Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 2. Berlin etc.: Springer-Verlag. XI, 470 p. DM 118 (1984)Google Scholar
  11. 11.
    Fulger, M.: The cones of effective cycles on projective bundles over curves. Math. Z. 269(1–2), 449–459 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hartshorne, R.: Ample vector bundles on curves. Nagoya Math. J. 43, 73–89 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jow, S.-Y.: Okounkov bodies and restricted volumes along very general curves. Adv. Math. 223(4), 1356–1371 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kaveh, K., Khovanskii, A.G.: Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. 176(2), 925–978 (2012)Google Scholar
  15. 15.
    Küronya, A., Lozovanu, V.: Positivity of line bundles and Newton-Okounkov bodies. arXiv:1506.06525 (2015)
  16. 16.
    Küronya, A., Lozovanu, V., Maclean, C.: Convex bodies appearing as Okounkov bodies of divisors. Adv. Math. 229(5), 2622–2639 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lazarsfeld, R.: Positivity in Algebraic Geometry, I & II, vol. 48 & 49. Springer, Berlin (2004)Google Scholar
  18. 18.
    Lesieutre, J.: The diminished base locus is not always closed. Compos. Math. 150(10), 1729–1741 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér. 42(5), 783–835 (2009)Google Scholar
  20. 20.
    Le Potier, J.: Lectures on Vector Bundles. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  21. 21.
    Manivel, L.: Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence. Société Mathématique de France, Paris (1998)zbMATHGoogle Scholar
  22. 22.
    Muñoz, R., Di Sciullo, F., L. E., : Solá Conde. On the existence of a weak Zariski decomposition on projectivized vector bundles. Geom. Dedicata 179, 287–301 (2015)Google Scholar
  23. 23.
    Miyaoka, Y.: The Chern classes and Kodaira dimension of a minimal variety. Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 449-476 (1987)Google Scholar
  24. 24.
    Nakayama, N.: Zariski-decomposition and abundance. Mathematical Society of Japan, Tokyo (2004)CrossRefzbMATHGoogle Scholar
  25. 25.
    Okounkov, A.: Brunn-Minkowski inequality for multiplicities. Invent. Math. 125(3), 405–411 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Okounkov, A.: Why would multiplicities be log-concave? In: The orbit method in geometry and physics. In honor of A. A. Kirillov. Papers from the international conference, Marseille, France, December 4–8, 2000, pp. 329–347. Birkhäuser, Boston (2003)Google Scholar
  27. 27.
    Ramanan, S., Ramanathan, A.: Some remarks on the instability flag. Tohoku Math. J. 2(36), 269–291 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wolfe, A.: Asymptotic Invariants of Graded Systems of Ideals and Linear Systems on Projective Bundles. Ph.D. Thesis, University of Michigan (2005)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut FourierUniv. Grenoble AlpesGrenobleFrance

Personalised recommendations