Lower bounds for Waldschmidt constants and Demailly’s conjecture for general and very general points

• Bisui, Sankhaneel ^[1] ; Nguyễn, Thái Thành ^[2]
  1. [1] Arizona State University

     Arizona State University

     Estados Unidos

     
  2. [2] University of Dayton

     University of Dayton

     City of Dayton, Estados Unidos

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 77, Fasc. 2, 2026, págs. 483-500
• Idioma: inglés
• DOI: 10.1007/s13348-025-00477-9
• Enlaces
  • Texto completo
• Resumen

  • We prove Demailly’s Conjecture concerning the lower bound for the Waldschmidt constant in terms of the initial degree of the second symbolic powers for any set of very general points in {\mathbb {P}}^N. We also discuss the Harbourne-Huneke Containment and the aforementioned Demailly’s Conjecture for general points and show the results for sufficiently many general points and general points in projective spaces with low dimensions.

• Referencias bibliográficas
  • Alexander, J.: Singularités imposables en positions générale à une hypersurface projective. Compos. Math. 68, 305–354 (1988)
  • Alexander, J., Hirschowitz, A.: Un lemme d’horace différentiel: application aux singularités hyperquartiques de \mathbb{P} ^5. J. Algebr....
  • Alexander, J., Hirschowitz, A.: La methode d’horace eclatée: application a l’interpolation en degré quatre. Invent. Math. 107, 585–602 (1992)
  • Alexander, J., Hirschowitz, A.: Polynomial interpolation in several variables. J. Algebr. Geom. 4, 201–222 (1995)
  • Bisui, S., Grifo, E., Hà, H.T., Nguyên, T.T.: Demailly’s conjecture and the containment problem. J. Pure Appl. Algebra 226(4), 106863 (2022)
  • Bisui, S., Grifo, E., Hà, H.T., Nguyên, T.T.: Chudnovsky’s conjecture and the stable Harbourne-Huneke containment. Trans. Amer. Math. Soc....
  • Bisui, S., Nguyên, T.T.: Chudnovsky’s conjecture and the stable Harbourne-Huneke containment for general points. J. Algebra 649, 245–269 (2024)
  • Bocci, C., Harbourne, B.: Comparing powers and symbolic powers of ideals. J. Algebr. Geom. 19(3), 399–417 (2010)
  • Bocci, C., Cooper, S.M., Harbourne, B.: Containment results for ideals of various configurations of points in \mathbb{P} ^N. J. Pure Appl....
  • Carlini, E., Hà, H.T., Harbourne, B., Van Tuyl, A.: Ideals of powers and powers of ideals: Intersecting Algebra, Geometry and Combinatorics,...
  • Chang, Y., Jow, S.: Demailly’s conjecture on Waldschmidt constants for sufficiently many very general points in \mathbb{P} ^n. J. Number Theory...
  • Chudnovsky, G.V.: Singular points on complex hypersurfaces and multidimensional Schwarz lemma. In: Séminaire de Théorie des Nombres, Paris...
  • Demailly, J.: Formules de Jensen en plusieurs variables et applications arithmétiques. Bull. Soc. Math. France 110(1), 75–102 (1982)
  • DiPasquale, M., Nguyên, T.T., Seceleanu, A.: Duality for asymptotic invariants of graded families. Adv. Math. 430, 109208 (2023)
  • DiPasquale, M., Villamizar, N.: A lower bound for splines on tetrahedral vertex stars. SIAM J. Appl. Algebra Geom. 5(2), 250–277 (2021)
  • Dumnicki, M., Harbourne, B., Szemberg, T., Tutaj-Gasińska, H.: Linear subspaces, symbolic powers and Nagata type conjectures. Adv. Math. 252,...
  • Dumnicki, M., Szemberg, T., Szpond, J.: Waldschmidt constants in projective spaces. J. Algebra 639(1), 1–22 (2024)
  • Dumnicki, M., Tutaj-Gasińska, H.: A containment result in \mathbb{P} ^n and the Chudnovsky conjecture. Proc. Amer. Math. Soc. 145(9), 3689–3694...
  • Dumnicki, M.: Symbolic powers of ideals of generic points in \mathbb{P} ^3. J. Pure Appl. Algebra 216(6), 1410–1417 (2012)
  • Dumnicki, M.: Containments of symbolic powers of ideals of generic points in \mathbb{P} ^3. Proc. Amer. Math. Soc. 143(2), 513–530 (2015)
  • Esnault, H., Viehweg, E.: Sur une minoration du degré d’hypersurfaces s’annulant en certains points. Math. Ann. 263(1), 75–86 (1983)
  • Evain, L.: On the postulation of sd fat points in pd. J. Algebra 285(2), 516–530 (2005)
  • Fouli, L., Mantero, P., Xie, Y.: Chudnovsky’s conjecture for very general points in \mathbb{P} ^N_k. J. Algebra 498, 211–227 (2018)
  • Gel’fand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants, and multidimensional determinants. Mathematics: Theory and Applications....
  • Geramita, A.V., Harbourne, B., Migliore, J.: Star configurations in \mathbb{P} ^n. J. Algebra 376, 279–299 (2013)
  • Harbourne, B., Huneke, C.: Are symbolic powers highly evolved? J. Ramanujan Math. Soc. 28A, 247–266 (2013)
  • Krull, W.: Parameterspezialisierung in Polynomringen. Arch. Math. 1, 56–64 (1948)
  • Lin, K., Shen, Y.: Symbolic powers of generalized star configurations hypersurfaces. J. Algebra 593, 193–216 (2022)
  • Malara, G., Szemberg, T., Szpond, J.: On a conjecture of Demailly and new bounds on Waldschmidt constants in \mathbb{P} ^N. J. Number Theory...
  • Moreau, J.-C.: Lemmes de schwarz en plusieurs variables et applications arithmétiques. In: Séminaire Pierre Lelong-Henri Skoda (Analyse)....
  • Nagel, U., Trok, B.: Interpolation and the weak Lefschetz property. Trans. Amer. Math. Soc. 372(12), 8849–8870 (2019)
  • Nguyên, T.T.: The initial degree of symbolic powers of Fermat-like ideals of planes and lines arrangements. Comm. Algebra 51(1), 29–45 (2023)
  • Nguyên, T.T.: Initial degree of symbolic powers of ideals of Fermat configurations of points. Rocky Mountain J. Math. 53(3), 859–874 (2023)
  • Tohaneanu, S., Xie, Y.: On the containment problem for fat points ideals. Proc. Amer. Math. Soc. 148(6), 2411–2419 (2020)
  • Waldschmidt, M.: Propriétés arithmétiques de fonctions de plusieurs variables. II. In: Séminaire Pierre Lelong (Analyse) année 1975/76, pp....