Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Maximal rank for schemes of small multiplicity by Évain's differential Horace method

Author: Joaquim Roé
Journal: Trans. Amer. Math. Soc. 366 (2014), 857-874
MSC (2010): Primary 14C20; Secondary 14H20, 14H50, 14D06
Published electronically: September 26, 2013
MathSciNet review: 3130319
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Hilbert function of the union of $ n$ general $ e$-fold points in the plane is maximal if $ n\ge 4e^2$ or $ n$ is a square. The Hilbert function of a union of $ A$, $ D$, $ E$ singularity schemes in general position is maximal in every degree $ >28$. The proofs use a computation of limits of families of linear systems whose special members acquire base divisors, an interesting problem in itself.

References [Enhancements On Off] (What's this?)

  • [1] J. Alexander and A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math. 140 (2000), no. 2, 303-325. MR 1756998 (2001i:14024),
  • [2] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932 (86h:14019)
  • [3] Anita Buckley and Marina Zompatori, Linear systems of plane curves with a composite number of base points of equal multiplicity, Trans. Amer. Math. Soc. 355 (2003), no. 2, 539-549 (electronic). MR 1932712 (2003j:14039),
  • [4] Eduardo Casas-Alvero, Singularities of plane curves, London Mathematical Society Lecture Note Series, vol. 276, Cambridge University Press, Cambridge, 2000. MR 1782072 (2003b:14035)
  • [5] P. Deligne, Intersections sur les surfaces régulières, SGA 7 II, Groupes de Monodromie En Géométrie Algébrique, Springer, LNM 340, 1973, pp. 1-38.
  • [6] Marcin Dumnicki, Expected term bases for generic multivariate Hermite interpolation, Appl. Algebra Engrg. Comm. Comput. 18 (2007), no. 5, 467-482. MR 2342565 (2008j:41003),
  • [7] L. Évain, Dimension des systèmes linéaires: une approche différentielle et combinatoire, preprint (1997),
  • [8] -, La fonction de Hilbert de la Réunion de 4$ ^h$ gros points génériques de $ {\mathbb{P}}^2$ de même multiplicité, J. Algebraic Geometry (1999), 787-796. MR 1703614 (2000e:13023)
  • [9] Laurent Evain, Computing limit linear series with infinitesimal methods, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 6, 1947-1974 (English, with English and French summaries). MR 2377892 (2009e:14044)
  • [10] G.-M. Greuel, C. Lossen, and E. Shustin, Singular algebraic curves, preprint, 409 pages, 2008.
  • [11] Brian Harbourne, Sandeep Holay, and Stephanie Fitchett, Resolutions of ideals of quasiuniform fat point subschemes of $ {\bf P}^2$, Trans. Amer. Math. Soc. 355 (2003), no. 2, 593-608 (electronic). MR 1932715 (2004c:13019),
  • [12] André Hirschowitz, La méthode d'Horace pour l'interpolation à plusieurs variables, Manuscripta Math. 50 (1985), 337-388 (French, with English summary). MR 784148 (86j:14013),
  • [13] Alexandre Junod, Hankel determinants and orthogonal polynomials, Expo. Math. 21 (2003), no. 1, 63-74. MR 1955218 (2004a:15009),
  • [14] Christian Radoux, Addition formulas for polynomials built on classical combinatorial sequences, Proceedings of the 8th International Congress on Computational and Applied Mathematics, ICCAM-98 (Leuven), 2000, pp. 471-477. MR 1747239 (2000m:05010),
  • [15] Joaquim Roé, Conditions imposed by tacnodes and cusps, Trans. Amer. Math. Soc. 353 (2001), no. 12, 4925-4948 (electronic). MR 1852087 (2002g:14007),
  • [16] Joaquim Roé, On the existence of plane curves with imposed multiple points, J. Pure Appl. Algebra 156 (2001), no. 1, 115-126. MR 1807019 (2001m:14050),
  • [17] Joaquim Roé, Maximal rank for planar singularities of multiplicity 2, J. Algebra 302 (2006), no. 1, 37-54. MR 2236593 (2007c:14022),

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14C20, 14H20, 14H50, 14D06

Retrieve articles in all journals with MSC (2010): 14C20, 14H20, 14H50, 14D06

Additional Information

Joaquim Roé
Affiliation: Department of Mathematics, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain

Received by editor(s): October 16, 2009
Received by editor(s) in revised form: May 14, 2012
Published electronically: September 26, 2013
Additional Notes: The author was partially supported by the Spanish Ministerio de ciecia e innovación grant MTM 2009-10359
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society