Dependent subsets of embedded projective varieties

Abstract

Let $X\subset {\mathbb{P}}^r$ be an integral and non-degenerate variety. Set $n:=\dim (X)$. Let $\rho (X{)}^{\prime \prime }$ 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 $\rho (X{)}^{\prime \prime }\ge \lceil (r+2)/2\rceil$ and that $\rho (X{)}^{\prime \prime }<2\lceil (r+1)/(n+1)\rceil$, unless either $n=r$ or $X$ is a rational normal curve.

References in corpus (2)

• Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture
• On the irreducibility of the Severi variety of nodal curves in a smooth surface