On the Secant Non-Defectivity of Integral Hypersurfaces of Projective Spaces of at Most Five Dimensions

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.