Dependent subsets of embedded projective varieties
Bull. Korean Math. Soc. 2020 Vol. 57, No. 4, 865-872
Published online October 24, 2019
Printed July 31, 2020
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 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)
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