On the infinitesimal Terracini Lemma

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 X \subseteq \mathcal P' is an irreducible, non-degenerate, projective complex variety of dimension n with r \geq 3n + 2 , such that the variety of osculating planes to curves in X has the expected dimension 3n and for every 0-dimensional, curvilinear scheme \gamma of length 3 contained in X the family of hyperplanes sections of X which are singular along \gamma has dimension larger that r-3(n+1) , then X is 2-secant defective.