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

Open Mathematics (2005)

- Volume: 3, Issue: 2, page 183-187
- ISSN: 2391-5455

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topEdoardo Ballico. "On the weak non-defectivity of veronese embeddings of projective spaces." Open Mathematics 3.2 (2005): 183-187. <http://eudml.org/doc/268907>.

@article{EdoardoBallico2005,

abstract = {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.},

author = {Edoardo Ballico},

journal = {Open Mathematics},

keywords = {14N05},

language = {eng},

number = {2},

pages = {183-187},

title = {On the weak non-defectivity of veronese embeddings of projective spaces},

url = {http://eudml.org/doc/268907},

volume = {3},

year = {2005},

}

TY - JOUR

AU - Edoardo Ballico

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

JO - Open Mathematics

PY - 2005

VL - 3

IS - 2

SP - 183

EP - 187

AB - 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.

LA - eng

KW - 14N05

UR - http://eudml.org/doc/268907

ER -

## References

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- [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 Zbl1045.14022
- [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. Zbl1078.14534
- [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. Zbl45.0239.01

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