Let X ⊂ Pn, where 3 ≤ n ≤ 5, be an irreducible hypersurface of degree d ≥ 2. Fix an integer t ≥ 3. If n = 5, assume t ≥ 4 and (t, d) ̸= (4, 2). Using the Differential Horace Lemma, we prove that OX(t) is not secant defective. For a fixed X, our proof uses induction on the integer t. The key points of our method are that for a fixed X, we only need the case of general linear hyperplane sections of X in lower-dimension projective spaces and that we do not use induction on d, allowing an interested reader to tackle a specific X for n > 5. We discuss (as open questions) possible extensions of some weaker forms of the theorem to the case where n > 5.
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