# On the infinitesimal Terracini Lemma

### Ciro Ciliberto

Università di Roma Tor Vergata, Italy

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## 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 $X \subseteq \mathcal P'$ 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.

## Cite this article

Ciro Ciliberto, On the infinitesimal Terracini Lemma. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 1, pp. 63–78

DOI 10.4171/RLM/926