Skip to main content
Log in

Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces

Revista Matemática Complutense Aims and scope Submit manuscript

Cite this article


We consider rational surfaces Z defined by divisorial valuations \(\nu \) of Hirzebruch surfaces. We introduce concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and global equivalent conditions. In particular we prove that, when \(\nu \) is non-positive at infinity, the extremal rays of the cone of curves of Z can be explicitly given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others


  1. Abhyankar, S.S., Moh, T.T.: Newton-Puiseux expansion and generalized Tschirnhausen transformation. I, II. J. Reine Angew. Math. 260, 47–83 (1973). ibid. 261 (1973), 29–54

    MathSciNet  MATH  Google Scholar 

  2. Beauville, A.: Complex Algebraic Surfaces. London Mathematical Society Student Texts, vol. 34, 2nd edn. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  3. Campillo, A.: Algebroid Curves in Positive Characteristic. Lecture Notes in Mathematics, vol. 613. Springer, Berlin (1980)

    Book  Google Scholar 

  4. Campillo, A., Piltant, O., Reguera, A.: Curves and divisors on surfaces associated to plane curves with one place at infinity. Proc. Lond. Math. Soc. 84, 559–580 (2002)

    Article  Google Scholar 

  5. Casas-Alvero, E.: Singularities of Plane Curves. London Mathematical Society Lecture Note Series, vol. 276. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  6. Ciliberto, C., Farnik, M., Küronya, A., Lozovanu, V., Roé, J., Shramov, C.: Newton–Okounkov bodies sprouting on the valuative tree. Rend. Circ. Mat. Palermo 2(66), 161–194 (2017)

    Article  MathSciNet  Google Scholar 

  7. Cutkosky, S.D., Ein, L., Lazarsfeld, R.: Positivity and complexity of ideal sheaves. Math. Ann. 321(2), 213–234 (2001)

    Article  MathSciNet  Google Scholar 

  8. de la Rosa-Navarro, B.L., Frías-Medina, J.B., Lahyane, M.: Rational surfaces with finitely generated Cox rings and very high Picard numbers. RACSAM 111, 297–306 (2017)

    Article  MathSciNet  Google Scholar 

  9. Delgado, F., Galindo, C., Núñez, A.: Saturation for valuations on two-dimensional regular local rings. Math. Z. 234, 519–550 (2000)

    Article  MathSciNet  Google Scholar 

  10. Demailly, J.P.: Singular Hermitian metrics on positive line bundles. Complex Algebraic Varieties (Bayreuth, 1990). Lecture Notes in Mathematics, vol. 1507, pp. 87–104. Springer, Berlin (1992)

    Chapter  Google Scholar 

  11. Dumnicki, M., Harbourne, B., Küronya, A., Roé, J., Szemberg, T.: Very general monomial valuations of \(\mathbb{P}^2\) and a Nagata type conjecture. Commun. Anal. Geom. 25, 125–161 (2017)

    Article  Google Scholar 

  12. Favre, C., Jonsson, M.: The Valuative Tree. Lecture Notes in Mathematics, vol. 1853. Springer, Berlin (2004)

    Book  Google Scholar 

  13. Favre, C., Jonsson, M.: Eigenvaluations. Ann. Sci. Éc. Norm. Sup. 40, 309–349 (2007)

    Article  MathSciNet  Google Scholar 

  14. Favre, C., Jonsson, M.: Dynamical compactifications of \(\mathbb{C}^2\). Ann. Math. 173, 211–248 (2011)

    Article  MathSciNet  Google Scholar 

  15. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)

    Book  Google Scholar 

  16. Galindo, C., Monserrat, F.: The cone of curves and the Cox ring of rational surfaces given by divisorial valuations. Adv. Math. 290, 1040–1061 (2016)

    Article  MathSciNet  Google Scholar 

  17. Galindo, C., Monserrat, F., Moyano-Fernández, J.: Minimal plane valuations. J. Algebraic Geom. 27, 751–783 (2018)

    Article  MathSciNet  Google Scholar 

  18. Galindo, C., Monserrat, F., Moyano-Fernández, J., Nickel, M.: Newton-Okounkov bodies of exceptional curve valuations. arXiv:1705.03948 (2017)

  19. Greco, S., Kiyek, K.: General elements of complete ideals and valuations centered at a two-dimensional regular local ring. In: Christensen, C., Sathaye, A., Sundaram, G., Bajaj, C. (eds.) Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), pp. 381–455. Springer, Berlin (2004)

    Chapter  Google Scholar 

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

    Book  Google Scholar 

  21. Iitaka, S.: On \(D\)-dimensions of algebraic varieties. J. Math. Soc. Jpn. 23, 356–373 (1971)

    Article  MathSciNet  Google Scholar 

  22. Jonsson, M.: Dynamics on Berkovich Spaces in Low Dimensions. Springer, Berlin (2015)

    Book  Google Scholar 

  23. Kaveh, K., Khovanskii, A.: Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. 176, 925–978 (2012)

    Article  MathSciNet  Google Scholar 

  24. Kollar, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  25. Moe, T.K.: Cuspidal curves on Hirzebruch surfaces. PhD thesis, Department of Mathematics, University of Oslo. (2013). Accessed 23 Nov 2018

  26. Mondal, P.: How to determine the sign of a valuation on \(\mathbb{C}[x, y]\). Mich. Math. J. 66, 227–244 (2017)

    Article  MathSciNet  Google Scholar 

  27. Okounkov, A.: Why would multiplicities be log-concave? In: Duval, C., Ovsienko, V., Guieu, L. (eds.) The Orbit Method in Geometry and Physics (Marseille, 2000). Progress in Mathematics, vol. 213. Birkhäuser, Boston (2003)

    Google Scholar 

  28. Reid, M.: Chapters on algebraic surfaces. In: Kollár, J. (ed.) Complex Algebraic Geometry (Park city, UT, 1993). IAS/Park City Lecture Notes Series, vol. 3, pp. 3–159. American Mathematical Society, Providence (1997)

    Chapter  Google Scholar 

  29. Spivakovsky, M.: Valuations in function fields of surfaces. Am. J. Math. 112, 107–156 (1990)

    Article  MathSciNet  Google Scholar 

  30. Teissier, B.: Valuations, deformations, and toric geometry. In: Kuhlmann, F.-Z., Kuhlmann, S., Marshall, M. (eds.) Valuation Theory and Its Applications, II (Saskatoon, SK, 1999). Fields Institute Communications, vol. 333. American Mathematical Society, Providence (1999)

    Google Scholar 

  31. Zariski, O., Samuel, P.: Commutative Algebra II. Vol. II. Graduate Texts in Mathematics, vol. 29. Springer, Berlin (1975)

    MATH  Google Scholar 

Download references


The authors thank M. Jonsson and W. Veys for valuable comments which helped to improve the paper.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Carlos Galindo.

Additional information

Publisher's Note

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

Partially supported by the Spanish Government Ministerio de Economía, Industria y Competitividad (MINECO), Grants MTM2015-65764-C3-2-P, MTM2016-81735-REDT, PGC2018-096446-B-C22 and BES-2016-076314, as well as by Universitat Jaume I, Grant UJI-B2018-10.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Galindo, C., Monserrat, F. & Moreno-Ávila, CJ. Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces. Rev Mat Complut 33, 349–372 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification