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

• Autorzy: Edoardo Ballico
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

Veronese variety; weakly defective variety; zero-dimensional scheme; double point; fat point; Veronese embedding; 14N05

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2005
• Tom: 3
• Numer: 2
• Strony: 183-187

Daty

wydano

2005-06-01

online

2005-06-01

Twórcy

autor

Edoardo Ballico

• University of Trento

Bibliografia

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• [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.
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• [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.

Identyfikatory

DOI

10.2478/BF02479194