Let $X$ be a smooth variety and let $x_{1},\ldots,x_{k}$ be general points. Let $T:=<T_{x_{1}}X,\ldots,T_{x_{k}}X>$ be the join of the tangent spaces at the points $x_{1},\ldots,x_{k}$. If we know that $T\cap X$ has only isolated singularities in an open neighborhood of $x_{1},\ldots,x_{k}$, is it true that $T\cap X$ is a local complete intersection?
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$\begingroup$ You're assuming a specified embedding of $X$ in some projective space, to take the join? $\endgroup$– Kevin CastoCommented Sep 26, 2019 at 17:31
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