Pseudo-effective cones of projective bundles and weak Zariski decomposition


We consider the projective bundle \(\mathbb {P}_X(E)\) over a smooth complex projective variety X, where E is a semistable bundle on X with \(c_2(\mathrm{End}(E)) =0\). We give a necessary and sufficient condition to get the equality \( \mathrm{Nef}^{\,1}(\mathbb {P}_X(E)) = \overline{\mathrm{Eff}}{}^{\,1}(\mathbb {P}_X(E))\) of nef cone and pseudo-effective cone of divisors in \(\mathbb {P}_X(E)\). As an application of our result, we show the equality of nef and pseudo-effective cones of divisors of projective bundles over some special varieties. In particular, we show that weak Zariski decomposition exists on these projective bundles. We also show that weak Zariski decomposition exists for fibre product \(\mathbb {P}_C(E)\,{\times }_C\,\mathbb {P}_C(E')\) over a smooth projective curve C. Finally, we show that a semistable bundle E of rank \(r\geqslant 2\) with \(c_2(\mathrm{End}(E)) = 0\) on a smooth complex projective surface of Picard number 1 is k-homogeneous, i.e., \(\overline{\mathrm{Eff}}{}^{\,k}(\mathbb {P}_X(E)) = \mathrm{Nef}^{\,k}(\mathbb {P}_{X}(E))\) for all \(1 \leqslant k < r\).

This is a preview of subscription content, access via your institution.


  1. 1.

    Birkar, C.: On existence of log minimal models and weak Zariski decompositions. Math. Ann. 354(2), 787–799 (2012)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Biswas, I., Bruzzo, U.: On semistable principal bundles over a complex projective manifold. Int. Math. Res. Not. IMRN 2008(12) (2008)

  3. 3.

    Biswas, I., Hogadi, A., Parameswaran, A.J.: Pseudo-effective cone of Grassmann bundles over a curve. Geom. Dedic. 172, 69–77 (2014)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Boucksom, S., Demailly, J.-P., Păun, 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)

    Article  Google Scholar 

  5. 5.

    Chen, D., Coskun, I.: Extremal higher codimension cycles on moduli spaces of curves. Proc. London Math. Soc. 111(1), 181–204 (2015)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Coskun, I., Lesieutre, J., Ottem, J.C.: Effective cones of cycles on blowups of projective space. Algebra Number Theory 10(9), 1983–2014 (2016)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cutkosky, S.D.: Zariski decomposition of divisors on algebraic varieties. Duke Math. J. 53(1), 149–156 (1986)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Debarre, O., Ein, L., Lazarsfeld, R., Voisin, C.: Pseudoeffective and nef classes on abelian varieties. Compositio Math. 147(6), 1793–1818 (2011)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Fujita, T.: On Zariski problem. Proc. Japan Acad. Ser. A Math. Sci. 55(3), 106–110 (1979)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Fujita, T.: Zariski decomposition and canonical rings of elliptic threefolds. J. Math. Soc. Japan 38(1), 19–37 (1986)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Fulger, M.: The cones of effective cycles on projective bundles over curves. Math. Z. 269(1–2), 449–459 (2011)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Fulton, W.: Intersection Theory. 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 2. Springer, Berlin (1998)

    Google Scholar 

  13. 13.

    Hartshorne, R.: Algebraic Geometry. Graduate Text in Mathematics, vol. 52. Springer, New York (1977)

    Google Scholar 

  14. 14.

    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, 2nd edn. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  15. 15.

    Karmakar, R., Misra, S.: Nef cone and Seshadri constants on products of projective bundles over curves. J. Ramanujan Math. Soc. 35(4), 317–325 (2020)

    Google Scholar 

  16. 16.

    Karmakar, R., Misra, S., Ray, N.: Nef and Pseudoeffective cones of product of projective bundles over a curve. Bull. Sci. Math. 151, 1–12 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Lazarsfeld, R.: Positivity in Algebraic Geometry, Vol. I. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48. Springer, Berlin (2007)

    Google Scholar 

  18. 18.

    Lazarsfeld, R.: Positivity in Algebraic Geometry, Vol. II. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 49. Springer, Berlin (2011)

    Google Scholar 

  19. 19.

    Lesieutre, J.: The diminished base locus is not always closed. Compositio Math. 150(10), 1729–1741 (2014)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Mehta, V.B., Nori, M.V.: Semistable sheaves on homogeneous spaces and abelian varieties. Proc. Indian Acad. Sci. Math. Sci. 93(1), 1–12 (1984)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Misra, S., Ray, N.: Nef cones of projective bundles over surfaces and Seshadri constants (2019). arXiv:1904.02335

  22. 22.

    Misra, S.: Stable Higgs bundles on ruled surfaces. Indian J. Pure Appl. Math. 51(2), 735–747 (2020)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Miyaoka, Y.: The Chern classes and Kodaira dimension of a minimal variety. In: Oda, T. (ed.) Algebraic Geometry, Sendai 1985, Advanced Studies in Pure Mathematics, vol. 10, pp. 449–476. North-Holland, Amsterdam (1987)

    Google Scholar 

  24. 24.

    Mukai, S.: Semi-homogeneous vector bundles on an Abelian variety. J. Math. Kyoto Univ. 18(2), 239–272 (1978)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Muñoz, R., Di Sciullo, F., Solá Conde, L.E.: On the existence of a weak Zariski decomposition on projectivized vector bundles. Geom. Dedicata 179, 287–301 (2015)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Nakayama, N.: Normalized tautological divisors of semi-stable vector bundles. Free resolutions of coordinate rings of projective varieties and related topics (Kyoto, 1998). Sūrikaisekikenkyūsho Kōkyūroku No. 1078, 167–173 (1999) (in Japanese)

  27. 27.

    Nakayama, N.: Zariski-Decomposition and Abundance. MSJ Memoirs, vol. 14. Mathematical Society of Japan, Tokyo (2004)

  28. 28.

    Prokhorov, Y.G.: On the Zariski decomposition problem. Proc. Steklov Inst. Math. 2003(1), 37–65 (2003)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Zariski, O.: The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface. Ann. Math. 76, 560–615 (1962)

    MathSciNet  Article  Google Scholar 

Download references


The author would like to thank Indranil Biswas and Omprokash Das for many useful discussions.

Author information



Corresponding author

Correspondence to Snehajit Misra.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported financially by a postdoctoral fellowship from TIFR, Mumbai under DAE, Government of India.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Misra, S. Pseudo-effective cones of projective bundles and weak Zariski decomposition. European Journal of Mathematics (2021).

Download citation


  • Nef cone
  • Pseudo-effective cone
  • Projective bundle
  • Semistability
  • Weak Zariski decomposition

Mathematics Subject Classification

  • 14J60
  • 14E25
  • 14N05
  • 14J40
  • 14E30
  • 14C17