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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.
We study the local behaviour of inflection points of families of plane curves in the
projective plane. We develop normal forms and versal deformation concepts for holomorphic
function germs which take into account
the inflection points of the fibres of . We give a classification of such function-
germs which is a projective analog of Arnold’s A,D,E classification. We compute the
versal deformation with respect to inflections of Morse function-germs.
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