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

Edoardo Ballico — 2005

Open Mathematics

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.

On projective degenerations of Veronese spaces

Edoardo Ballico — 1996

Banach Center Publications

Here we give several examples of projective degenerations of subvarieties of t . The more important case considered here is the d-ple Veronese embedding of n ; we will show how to degenerate it to the union of d n n-dimensional linear subspaces of t ; t : = ( n + d ) / ( n ! d ! ) - 1 and the union of scrolls. Other cases considered in this paper are essentially projective bundles over important varieties. The key tool for the degenerations is a general method due to Moishezon. We will give elsewhere several applications to postulation...

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