Initial degree and Waldschmidt constant of zero schemes and properties
Abstract
We give a short survey about evaluation of initial degree and Waldschmidt constant of zero schemes in projective space. We show main and updated results, some related conjectures and new computations of initial degree and Waldschmidt constant
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References
[1] C. Bocci and B. Harbourne, Comparing powers and symbolic power of ideals, J. Algebraic Geom. 19 ( 2010), 399–417.
[2] Bocci, C., Cooper, S., Guardo, E. et al. (2016), The Waldschmidt constant for squarefree monomial ideals, J. Algebr. Comb. 44, 875–904 .
[3] G. V. Chudnovsky, Singular points on complex hypersurfaces and multidimensional Schwarz Lemma, Seminaire de Théorie des Nombres, Paris 1979-80, Séminaire Delange-Pisot-Poitou, Progress in Math vol. 12, M-J Bertin, editor, Birkhauser, Boston-Basel-Stutgart (1981).
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[6] M. Dumnicki, H.T. Gasinska, A containment result in Pn and the Chudnovsky conjecture, Proc. Amer. Math. Soc. 145, (2017), 3689-3694.
[7] L. Ein, R. Lazarsfeld, and K.E. Smith, Uniform behavior of symbolic powers of ideals, Invent. Math., 144 (2001), 241-252.
[8] H. Esnault and E. Viehweg, Sur une minoration du degré d’hypersurfaces s’annulant en certains points, Math. Ann. 263 (1983), no. 1, 75-86.
[9] L. Evain, On the postulation of sd fat points in Pd , J. Algebra 285 (2005), 516-530.
[10] L. Fouli, P. Mantero, Y. Xie, Chudnovsky’s Conjecture for very general points in Pnk , J. of Algebra, 498 (2018), 211-227.
[11] R. Harshorne, Algebraic Geometry, Spinger-Verlag, (1977)
[12] B. Harbourne and C. Huneke Are symbolic powers highly evolved?, J. Ramanujan Math. Soc. 28A (2013), 247-266.
[13] M. Hochster and C. Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), no. 2, 349-369
[14] M. Janssen, (2013) Symbolic Powers of Ideals in k[PN], Dissertations, Theses and Student Research Papers in Mathematics. 41, Lincoln, Nebraska.
[15] M. Nagata, On the 14-th problem of Hilbert, Amer. J. Math. 33 (1959), 766-772.
[16] M. Nagata, On rational surfaces, II, Mem. College Sci. Univ. Kyoto Ser. A Math. 33 (1960), 271-293.
[17] A. Iarrobino. Inverse system of a symbolic power III: thin algebras and fat points, Compositio Math. 108, (1997), 319–356.
[18] H. Skoda, Estimations L2 pour l’operateur ^ et applications arithmetiques, in: Seminaire P. Lelong (Analyse), 1975/76, Lecture Notes in Mathematics 578, Springer, 1977, 314-323.
[19] N.C. Tu, D.T. Hiep, L.N.Long, V.Thanh, Waldschmidt constant of some sets of points in projective plane, (2020), (preprint).
[20] M. Waldschmidt, Propriétés arithmétiques de fonctions de plusieurs variables II, Séminaire P. Lelong (Analyse), 1975-76, Lecture Notes Math. 578, Springer-Verlag, 1977, 108-135.
[21] M. Waldschmidt, Nombres transcendants et groupes algébriques, Astérisque 69/70, Socéte Mathématiqué de France, 1979
[2] Bocci, C., Cooper, S., Guardo, E. et al. (2016), The Waldschmidt constant for squarefree monomial ideals, J. Algebr. Comb. 44, 875–904 .
[3] G. V. Chudnovsky, Singular points on complex hypersurfaces and multidimensional Schwarz Lemma, Seminaire de Théorie des Nombres, Paris 1979-80, Séminaire Delange-Pisot-Poitou, Progress in Math vol. 12, M-J Bertin, editor, Birkhauser, Boston-Basel-Stutgart (1981).
[4] S. M. Cooper and B. Harbourne, Regina Lectures On Fat Points, Springer Proceedings in Mathematics and Statistics, Vol. 76 (2014), ISBN: 978-1-4939-0625-3.
[5] M. Dumnicki, Symbolic powers of ideals of generic points in Pn, J. Pure Appl. Algebra 216 (2012), 1410-1417.
[6] M. Dumnicki, H.T. Gasinska, A containment result in Pn and the Chudnovsky conjecture, Proc. Amer. Math. Soc. 145, (2017), 3689-3694.
[7] L. Ein, R. Lazarsfeld, and K.E. Smith, Uniform behavior of symbolic powers of ideals, Invent. Math., 144 (2001), 241-252.
[8] H. Esnault and E. Viehweg, Sur une minoration du degré d’hypersurfaces s’annulant en certains points, Math. Ann. 263 (1983), no. 1, 75-86.
[9] L. Evain, On the postulation of sd fat points in Pd , J. Algebra 285 (2005), 516-530.
[10] L. Fouli, P. Mantero, Y. Xie, Chudnovsky’s Conjecture for very general points in Pnk , J. of Algebra, 498 (2018), 211-227.
[11] R. Harshorne, Algebraic Geometry, Spinger-Verlag, (1977)
[12] B. Harbourne and C. Huneke Are symbolic powers highly evolved?, J. Ramanujan Math. Soc. 28A (2013), 247-266.
[13] M. Hochster and C. Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), no. 2, 349-369
[14] M. Janssen, (2013) Symbolic Powers of Ideals in k[PN], Dissertations, Theses and Student Research Papers in Mathematics. 41, Lincoln, Nebraska.
[15] M. Nagata, On the 14-th problem of Hilbert, Amer. J. Math. 33 (1959), 766-772.
[16] M. Nagata, On rational surfaces, II, Mem. College Sci. Univ. Kyoto Ser. A Math. 33 (1960), 271-293.
[17] A. Iarrobino. Inverse system of a symbolic power III: thin algebras and fat points, Compositio Math. 108, (1997), 319–356.
[18] H. Skoda, Estimations L2 pour l’operateur ^ et applications arithmetiques, in: Seminaire P. Lelong (Analyse), 1975/76, Lecture Notes in Mathematics 578, Springer, 1977, 314-323.
[19] N.C. Tu, D.T. Hiep, L.N.Long, V.Thanh, Waldschmidt constant of some sets of points in projective plane, (2020), (preprint).
[20] M. Waldschmidt, Propriétés arithmétiques de fonctions de plusieurs variables II, Séminaire P. Lelong (Analyse), 1975-76, Lecture Notes Math. 578, Springer-Verlag, 1977, 108-135.
[21] M. Waldschmidt, Nombres transcendants et groupes algébriques, Astérisque 69/70, Socéte Mathématiqué de France, 1979
Published
2020-12-29
How to Cite
NGUYEN, Tu Chanh.
Initial degree and Waldschmidt constant of zero schemes and properties.
Journal of Science and Technology: Issue on Information and Communications Technology, [S.l.], v. 18, n. 12.2, p. 15-18, dec. 2020.
ISSN 1859-1531.
Available at: <http://ict.jst.udn.vn/index.php/jst/article/view/111>. Date accessed: 02 may 2024.
doi: https://doi.org/10.31130/ict-ud.2020.111.
Section
Articles
