Skip to main content

All Fat Point Subschemes in \({\mathbb {P}}^2\) with the Waldschmidt Constant Less than 5 / 2


Let \({\mathscr {A}}=m_1p_1+ \cdots +m_np_n\) be a fat point subscheme of \({\mathbb {P}}^2\), and let \(I({\mathscr {A}})\), which is called a fat point ideal, be its corresponding ideal in \({\mathbb {K}}[{\mathbb {P}}^2]\). In this note, we identify those fat point ideals in \({\mathbb {K}} [{\mathbb {P}}^2]\) for which their Waldschmidt constants are less than 5 / 2.

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

Fig. 1
Fig. 2
Fig. 3


  1. 1.

    Bocci, C., Harbourne, B.: Comparing powers and symbolic powers of ideals. J. Algebraic Geom. 19(3), 399–417 (2010)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Chang, Y., Jow, S.: Demailly’s conjecture on Waldschmodt constants for sufficiently many very general points in \({\mathbb{P}}^N\) (2019). (preprint). arXiv:1903.05824v1

  3. 3.

    Ciliberto, C., Miranda, R.: Nagata’s conjecture for a square or nearly-square number of points. Ric. Mat. 55(1), 71–78 (2006)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Demailly, J.P.: Formules de Jensen en plusieurs variables et applications arithmétiques. Bull. Soc. Math. France 110(1), 75–102 (1992)

    MATH  Google Scholar 

  5. 5.

    Dumnicki, M., Szemberg, T., Tutaj-Gasińska, H.: Symbolic powers of planar point configurations. J. Pure Appl. Algebra 217(6), 1026–1036 (2013)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Dumnicki, M., Szemberg, T., Tutaj-Gasińska, H.: Symbolic powers of planar point configurations II. J. Pure Appl. Alg 220, 2001–2016 (2016)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Evain, L.: it Computing limit linear series with infinitesimal methods. Ann. Inst. Fourier 57, 307–327 (2007)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Farnik, Ł., Gwoździewicz, J., Hejmej, B., Lampa-Baczyńska, M., Malara, G., Szpond, J.: Initial sequence and Waldschmidt constants of planar point configurations, internat. J. Algebra Comput. 27(6), 717–729 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Haghighi, H., Mosakhani, M., Fashami, M.Zaman: Resurgence and Waldschmidt constant of the ideal of fat almost collinear subscheme in \({\mathbb{P}}^2\). Ann. Univ. Paedagog. Crac. Stud. Math. 17, 59–65 (2018)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Haghighi, H., Mosakhani, M.: All symbolic powers and ordinary powers of the defining ideal of a fat nearly-complete intersection are equal. J. Algebra Appl. 18(8), 1950142 (2019). 9

    MathSciNet  Article  Google Scholar 

  11. 11.

    Harbourne, B.: The atlas of Waldschmidt constants (2019).

  12. 12.

    Harbourne, B., Huneke, C.: Are symbolic powers highly evolved. J. Ramunajan Math. Soc. 28A, 247–266 (2013)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Hirschowitz, A.: La méthod d’Horace pour l’interpolation à plusieurs variables. Manus. Math. 50, 337–388 (1985)

    Article  Google Scholar 

  14. 14.

    Malara, G., Szemberg, T., Szpond, J.: On a conjecture of Demailly and new bounds on Waldschmidt constants in \({\mathbb{P}}^N\). J. Number Theory 189, 211–219 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Moreau, J.C.: Lemmes de Schwartz en plusieurs variables et applications arithmétique, in Séminaire Pierre Leleng-Henri Skoda (Analyse). Années 1978/79 (French). In: Lecture Notes in Math., vol. 822, pp. 174–190. Springer, Berlin (1980)

  16. 16.

    Mosakhani, M., Haghighi, H.: On the configurations of points in \({\mathbb{P}}^2\) with the Waldschmidt constant equal to two. J. Pure Appl. Algebra 220(12), 3821–3825 (2016)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Nagata, M.: On the 14th problem of Hilbert. Amer. J. Math. 81, 766–772 (1959)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Reed, D.: On a problem of Zariski. Illinois J. Math. 2, 145–149 (1958)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Schenzel, P.: Filtrations and Noetherian symbolic blow-up rings. Proc. Amer. Math. Soc. 102(4), 817–822 (1998)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Waldschmidt, M.: Properiétés arithmétiques de fonctions de plusieurs variables II. Lecture Notes Math. 578, 108–135 (1977)

    Article  Google Scholar 

  21. 21.

    Zariski, O., Samuel, P.: Commutative algebra, vol. II. Reprint of the 1960 edition. Graduate Texts in Mathematics, vol. 29, X+414. Springer, Heidelberg (1975)

Download references


We would like to thank the anonymous referee for her/his careful reading of this manuscript, valuable suggestions and making helpful remarks. These all helped to improve the manuscript. This paper was prepared based on a research project supported by K.N. Toosi University of Technology research council and Iran National Science Foundation (INSF) Grant No. 97008366.

Author information



Corresponding author

Correspondence to Hassan Haghighi.

Additional information

Publisher's Note

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

Communicated by Siamak Yassemi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Haghighi, H., Mosakhani, M. All Fat Point Subschemes in \({\mathbb {P}}^2\) with the Waldschmidt Constant Less than 5 / 2. Bull. Malays. Math. Sci. Soc. 43, 3221–3228 (2020).

Download citation


  • Configuration of points
  • Star configuration
  • Symbolic power
  • Waldschmidt constant
  • Fat points

Mathematics Subject Classification

  • Primary 14N20
  • 13A02
  • Secondary 14N05
  • 13F20