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2005 | 3 | 2 | 183-187
Tytuł artykułu

On the weak non-defectivity of veronese embeddings of projective spaces

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EN
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EN
Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<(nn+x). Here we prove that the order x Veronese embedding ofP n is not weakly (k−1)-defective, i.e. for a general S⊃P n such that #(S) = k+1 the projective space | I 2S (x)| of all degree t hypersurfaces ofP n singular at each point of S has dimension (n/n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I 2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S.
Wydawca
Czasopismo
Rocznik
Tom
3
Numer
2
Strony
183-187
Opis fizyczny
Daty
wydano
2005-06-01
online
2005-06-01
Twórcy
Bibliografia
  • [1] J. Alexander: “Singularités imposables en position générale aux hypersurfaces de ℙn”, Compositio Math., Vol. 68, (1988), pp. 305–354.
  • [2] J. Alexander and A. Hirschowitz: “Un lemme d'Horace différentiel: application aux singularité hyperquartiques de ℙ5”, J. Algebraic Geom., Vol. 1, (1992), pp. 411–426.
  • [3] J. Alexander and A. Hirschowitz: “La méthode d'Horace éclaté: application à l'interpolation en degré quatre”, Invent. Math., Vol. 107, (1992), pp. 585–602. http://dx.doi.org/10.1007/BF01231903
  • [4] J. Alexander and A. Hirschowitz: “Polynomial interpolation in several variables”, J. Algerraic Geom., Vol. 4, (1995), pp. 201–222.
  • [5] J. Alexander and A. Hirschowitz: “An asymptotic vanishing theorem for generic unions of multiple points”, Invent. Math., Vol. 140, (2000), pp. 303–325. http://dx.doi.org/10.1007/s002220000053
  • [6] K. Chandler: “A brief proof of a maximal rank theorem for generic double points in projective space”, Trans. Amer. Math. Soc., Vol. 353(5), (2000), pp. 1907–1920. http://dx.doi.org/10.1090/S0002-9947-00-02732-X
  • [7] L. Chiantini and C. Ciliberto: “Weakly defective varieties”, Trans. Amer. Math. Soc., Vol. 454(1), (2002), pp. 151–178. http://dx.doi.org/10.1090/S0002-9947-01-02810-0
  • [8] C. Ciliberto: “Geometric aspects of polynomial interpolation in more variables and of Waring's problem”, In European Congress of Mathematics (Barcelona, 2000), Progress in Math., Vol. 201, Birkhäuser, Basel 2001, pp. 289–316.
  • [9] C. Ciliberto and A. Hirschowitz: “Hypercubique de ℙ4 avec sept points singulieres génériques”, C. R. Acad. Sci. Paris, Vol. 313(1), (1991), pp. 135–137.
  • [10] M. Mella: Singularities of linear systems and the Waring problem, arXiv mathAG/0406288.
  • [11] A. Terracini: “Sulla rappresentazione delle coppie di forme ternarie mediante somme di potenze di forme lineari”, Ann. Mat. Pura e Appl., Vol. 24, (1915), pp. 91–100.
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_BF02479194
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