Bulletin of the
Korean Mathematical Society BKMS

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 varietys $X$-ranks zero-dimensional schemes variety with only one ordinary double points OADP

MSC numbers: 14N05

Supported by: The author was partially supported by MIUR and GNSAGA of INdAM (Italy)