On the infinitesimal Terracini Lemma

Abstract

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 $X\subseteq {\mathcal{P}}^{\prime }$ is an irreducible, non-degenerate, projective complex variety of dimension $n$ with $r\ge 3n+2$, 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.