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 of the union of points of the same multiplicity in the plane up to degree .
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 for all . As a second application, we compute collisions of fat points in the plane.
Alexander et Hirschowitz ont calculé la fonction de Hilbert d’une réunion générique de gros points du plan projectif sous l’hypothèse que le nombre de gros points est très supérieur à leur multiplicité. Leur méthode est basée sur un lemme permettant le calcul d’un système linéaire limite lorsque les gros points se spécialisent sur un diviseur.
Nagata avait auparavant calculé d’autres fonctions de Hilbert. Lors de la construction de son contre-exemple au quatorzième problème de Hilbert, Nagata a déterminé la fonction de Hilbert d’une réunion de gros points de même multiplicité lorsque .
On introduit une nouvelle méthode de calcul de systèmes linéaires limites, qui généralise le résultat de Alexander et Hirschowitz. Notre principale application est de compléter le résultat de Nagata : nous calculons pour tout . Comme autre application, nous décrivons des collisions de gros points dans le plan projectif.
Keywords: Fat point, Hilbert function, Nagata, curve, singularities
Mot clés : gros point, fonction de Hilbert, Nagata, courbe, singularités
Evain, Laurent 1
@article{AIF_2007__57_6_1947_0, author = {Evain, Laurent}, title = {Computing limit linear series with infinitesimal methods}, journal = {Annales de l'Institut Fourier}, pages = {1947--1974}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {6}, year = {2007}, doi = {10.5802/aif.2319}, mrnumber = {2377892}, zbl = {1134.14020}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2319/} }
TY - JOUR AU - Evain, Laurent TI - Computing limit linear series with infinitesimal methods JO - Annales de l'Institut Fourier PY - 2007 SP - 1947 EP - 1974 VL - 57 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2319/ DO - 10.5802/aif.2319 LA - en ID - AIF_2007__57_6_1947_0 ER -
%0 Journal Article %A Evain, Laurent %T Computing limit linear series with infinitesimal methods %J Annales de l'Institut Fourier %D 2007 %P 1947-1974 %V 57 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2319/ %R 10.5802/aif.2319 %G en %F AIF_2007__57_6_1947_0
Evain, Laurent. Computing limit linear series with infinitesimal methods. Annales de l'Institut Fourier, Volume 57 (2007) no. 6, pp. 1947-1974. doi : 10.5802/aif.2319. https://aif.centre-mersenne.org/articles/10.5802/aif.2319/
[1] An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math., Volume 140 (2000) no. 2, pp. 303-325 | DOI | MR | Zbl
[2] Infinitely near imposed singularities and singularities of polar curves, Math. Ann., Volume 287 (1990) no. 3, pp. 429-454 | DOI | MR | Zbl
[3] Geometric aspects of polynomial interpolation in more variables and of Waring’s problem, European Congress of Mathematics, Vol. I (Barcelona, 2000) (Progr. Math.), Volume 201, Birkhäuser, Basel, 2001, pp. 289-316 | Zbl
[4] On the symmetric product of a curve with general moduli, Geometriae Dedicata, Volume 78 (1999), pp. 327-343 | DOI | MR | Zbl
[5] Matching conditions for degenerating plane curves and applications (To appear) | Zbl
[6] Degenerations of planar linear systems, J. Reine Angew. Math., Volume 501 (1998), pp. 191-220 | MR | Zbl
[7] Calculs de dimensions de systèmes linéaires de courbes planes par collisions de gros points, C. R. Acad. Sci. Paris Sér. I Math., Volume 325 (1997) no. 12, pp. 1305-1308 | DOI | MR | Zbl
[8] Collisions de trois gros points sur une surface algébrique, PhD., Nice (1997) (Ph. D. Thesis)
[9] La fonction de Hilbert de la réunion de gros points génériques de de même multiplicité, J. Algebraic Geom., Volume 8 (1999) no. 4, pp. 787-796 | MR | Zbl
[10] Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. (1960) no. 4, pp. 228 | Numdam | MR | Zbl
[11] Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1995, pp. Exp. No. 221, 249-276 | Numdam | Zbl
[12] La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Mathematica, Volume 50 (1985), pp. 337-388 | DOI | Zbl
[13] Symplectic packings and algebraic geometry, Invent. Math., Volume 115 (1994) no. 3, pp. 405-434 (With an appendix by Yael Karshon) | DOI | MR | Zbl
[14] On Rational Surfaces, II, Memoirs of the College of Science, University of Kyoto, Volume XXXIII (1960) no. 2, pp. 271-293 | MR | Zbl
[15] On the fourteenth problem of Hilbert, Proc. Internat. Congress Math. 1958, Cambridge Univ. Press, New York, 1960, pp. 459-462 | MR | Zbl
[16] Collisions of three fat points on an algebraic surface, Prépublication 412, Univ. Nice (1995), pp. 1-7
[17] Ample line bundles on smooth surfaces, J. reine angew. Math., Volume 469 (1995), pp. 199-209 | DOI | MR | Zbl
Cited by Sources: