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 Dependent subsets of embedded projective varieties Bull. Korean Math. Soc.Published online October 24, 2019 Edoardo Ballico University of Trento Abstract : Let $X\subset \mathbb {P}^r$ be an integral and non-degenerate variety. Set $n:= \dim (X)$. Let $\rho (X)''$ be the maximal integer such that every zero-dimensional scheme $Z\subset X$ smoothable in $X$ is linearly independent. We prove that $X$ is linearly normal if $\rho (X)''\ge \lceil (r+2)/2\rceil$ and that $\rho (X)'' < 2\lceil (r+1)/(n+1)\rceil$, unless either $n=r$ or $X$ is a rational normal curve. Keywords : secant variety; $X$-rank; zero-dimensional scheme; variety with only one ordinary double point; OADP MSC numbers : 14N05 Full-Text :