Abstract
Fix integers n, x, k such that nâ„3, k>0, xâ„4, (n, x)â (3, 4) and k(n+1)<( n n+x ). Here we prove that the order x Veronese embedding ofP n is not weakly (kâ1)-defective, i.e. for a general SâP 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 )â1â k(n+1) (proved by Alexander and Hirschowitz) and a general Fâ| I 2S (x)| has an ordinary double point at each Pâ S and Sing (F)=S.
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