Computing limit linear series with infinitesimal methods
Annales de l'Institut Fourier, Volume 57 (2007) no. 6, p. 1947-1974
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.
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 H(d) d’une réunion de k 2 gros points de même multiplicité m lorsque dkm.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 H(d) pour tout d. Comme autre application, nous décrivons des collisions de gros points dans le plan projectif.
DOI : https://doi.org/10.5802/aif.2319
Classification:  14H20,  14H50,  14B05,  14B10,  14B20
Keywords: Fat point, Hilbert function, Nagata, curve, singularities
@article{AIF_2007__57_6_1947_0,
     author = {Evain, Laurent},
     title = {Computing limit linear series with infinitesimal methods},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {6},
     year = {2007},
     pages = {1947-1974},
     doi = {10.5802/aif.2319},
     zbl = {1134.14020},
     mrnumber = {2377892},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2007__57_6_1947_0}
}
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/item/AIF_2007__57_6_1947_0/

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