On the infinitesimal Terracini lemma

TL;DR

This paper establishes an infinitesimal version of the classical Terracini Lemma for 3-secant planes, providing conditions under which a variety is 2-secant defective, thus advancing understanding of secant varieties in algebraic geometry.

Contribution

It proves an infinitesimal version of the Terracini Lemma for 3-secant planes, linking osculating planes and secant defectiveness under specific geometric conditions.

Findings

Variety of osculating planes has expected dimension 3n.

Hyperplanes singular along certain schemes have larger than expected dimension.

Under conditions, the variety is proven to be 2-secant defective.

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 $X\subseteq \PP^r$ 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.