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In this paper we prove an infinitesimal version of the classical Terracini Lemma for 3-secant planes to a variety. Precisely we prove that if is an irreducible, non-degenerate, projective complex variety of dimension with , such that the variety of osculating planes to curves in has the expected dimension and for every 0-dimensional, curvilinear scheme of length 3 contained in the family of hyperplanes sections of which are singular along has dimension larger that , then is 2-secant defective.
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Ciro Ciliberto, On the infinitesimal Terracini Lemma. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 1, pp. 63–78DOI 10.4171/RLM/926