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.

中文翻译：

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关于无穷小Terracini引理

在本文中，我们证明了用于3割平面的经典Terracini引理的无穷小版本。确切地说，我们证明如果$ X \ subseteq \ mathcal P'$是尺寸为$ n $且具有$ r \ geq 3n + 2 $的不可约的，非退化的射影复杂变体，从而使切合平面的各种曲线变为$ X $的预期维数为$ 3n $，对于$ X $中长度为3的每个0维曲线方案$ \ gamma $，$ X $的超平面部分族沿着$ \ gamma $奇异的维数大于$ r-3（n + 1）$，则$ X $是2割线缺陷。