Computing limit linear series with infinitesimal methods

Laurent Evain[1]

  • [1] Université d’Angers Faculté des Sciences Département de maths 2, Boulevard Lavoisier 49045 Angers Cedex 01 (France)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 6, page 1947-1974
  • ISSN: 0373-0956

Abstract

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Alexander and Hirschowitz determined the Hilbert function of a generic union of fat points in a projective space when the number of fat points is much bigger than the greatest multiplicity of the fat points. Their method is based on a lemma which determines the limit of a linear system depending on fat points approaching a divisor.Other Hilbert functions were computed previously by Nagata. In connection with his counter-example to Hilbert’s fourteenth problem, Nagata determined the Hilbert function H ( d ) of the union of k 2 points of the same multiplicity m in the plane up to degree d = k m .We introduce a new method to determine limits of linear systems. This generalizes the result by Alexander and Hirschowitz. Our main application of this method is the conclusion of the work initiated by Nagata: we compute H ( d ) for all d . As a second application, we compute collisions of fat points in the plane.

How to cite

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Evain, Laurent. "Computing limit linear series with infinitesimal methods." Annales de l’institut Fourier 57.6 (2007): 1947-1974. <http://eudml.org/doc/10283>.

@article{Evain2007,
abstract = {Alexander and Hirschowitz determined the Hilbert function of a generic union of fat points in a projective space when the number of fat points is much bigger than the greatest multiplicity of the fat points. Their method is based on a lemma which determines the limit of a linear system depending on fat points approaching a divisor.Other Hilbert functions were computed previously by Nagata. In connection with his counter-example to Hilbert’s fourteenth problem, Nagata determined the Hilbert function $H(d)$ of the union of $k^2$ points of the same multiplicity $m$ in the plane up to degree $d=km$.We introduce a new method to determine limits of linear systems. This generalizes the result by Alexander and Hirschowitz. Our main application of this method is the conclusion of the work initiated by Nagata: we compute $H(d)$ for all $d$. As a second application, we compute collisions of fat points in the plane.},
affiliation = {Université d’Angers Faculté des Sciences Département de maths 2, Boulevard Lavoisier 49045 Angers Cedex 01 (France)},
author = {Evain, Laurent},
journal = {Annales de l’institut Fourier},
keywords = {Fat point; Hilbert function; Nagata; curve; singularities; fat point; nagata},
language = {eng},
number = {6},
pages = {1947-1974},
publisher = {Association des Annales de l’institut Fourier},
title = {Computing limit linear series with infinitesimal methods},
url = {http://eudml.org/doc/10283},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Evain, Laurent
TI - Computing limit linear series with infinitesimal methods
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 6
SP - 1947
EP - 1974
AB - Alexander and Hirschowitz determined the Hilbert function of a generic union of fat points in a projective space when the number of fat points is much bigger than the greatest multiplicity of the fat points. Their method is based on a lemma which determines the limit of a linear system depending on fat points approaching a divisor.Other Hilbert functions were computed previously by Nagata. In connection with his counter-example to Hilbert’s fourteenth problem, Nagata determined the Hilbert function $H(d)$ of the union of $k^2$ points of the same multiplicity $m$ in the plane up to degree $d=km$.We introduce a new method to determine limits of linear systems. This generalizes the result by Alexander and Hirschowitz. Our main application of this method is the conclusion of the work initiated by Nagata: we compute $H(d)$ for all $d$. As a second application, we compute collisions of fat points in the plane.
LA - eng
KW - Fat point; Hilbert function; Nagata; curve; singularities; fat point; nagata
UR - http://eudml.org/doc/10283
ER -

References

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  1. J. Alexander, A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math. 140 (2000), 303-325 Zbl0973.14026MR1756998
  2. E. Casas-Alvero, Infinitely near imposed singularities and singularities of polar curves, Math. Ann. 287 (1990), 429-454 Zbl0675.14009MR1060685
  3. C. Ciliberto, Geometric aspects of polynomial interpolation in more variables and of Waring’s problem, European Congress of Mathematics, Vol. I (Barcelona, 2000) 201 (2001), 289-316, Birkhäuser, Basel Zbl1078.14534
  4. C. Ciliberto, A. Kouvidakis, On the symmetric product of a curve with general moduli, Geometriae Dedicata 78 (1999), 327-343 Zbl0967.14021MR1725369
  5. C. Ciliberto, R. Miranda, Matching conditions for degenerating plane curves and applications Zbl1109.14009
  6. C. Ciliberto, R. Miranda, Degenerations of planar linear systems, J. Reine Angew. Math. 501 (1998), 191-220 Zbl0943.14002MR1637857
  7. L. Evain, Calculs de dimensions de systèmes linéaires de courbes planes par collisions de gros points, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), 1305-1308 Zbl0905.14005MR1490419
  8. L. Evain, Collisions de trois gros points sur une surface algébrique, (1997) 
  9. L. Evain, La fonction de Hilbert de la réunion de 4 h gros points génériques de P 2 de même multiplicité, J. Algebraic Geom. 8 (1999), 787-796 Zbl0953.14027MR1703614
  10. A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. (1960) Zbl0118.36206MR217083
  11. A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6 (1995), Exp. No. 221, 249-276, Soc. Math. France, Paris Zbl0236.14003
  12. A. Hirschowitz, La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Mathematica 50 (1985), 337-388 Zbl0571.14002
  13. D. McDuff, L. Polterovich, Symplectic packings and algebraic geometry, Invent. Math. 115 (1994), 405-434 Zbl0833.53028MR1262938
  14. M. Nagata, On Rational Surfaces, II, Memoirs of the College of Science, University of Kyoto XXXIII (1960), 271-293 Zbl0100.16801MR126444
  15. M. Nagata, On the fourteenth problem of Hilbert, Proc. Internat. Congress Math. 1958 (1960), 459-462, Cambridge Univ. Press, New York Zbl0127.26302MR116056
  16. C. Walter, Collisions of three fat points on an algebraic surface, Prépublication 412, Univ. Nice (1995), 1-7 
  17. G. Xu, Ample line bundles on smooth surfaces, J. reine angew. Math. 469 (1995), 199-209 Zbl0833.14028MR1363830

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