# Dependent subsets of embedded projective varieties

## TL;DR

This paper investigates the properties of embedded projective varieties, establishing bounds on the maximal size of zero-dimensional schemes that are linearly independent within the variety, with implications for linear normality.

## Contribution

It provides new bounds on the maximal size of zero-dimensional schemes that are smoothable and linearly independent in embedded projective varieties, linking these bounds to the variety's dimension and embedding dimension.

## Key findings

- If $
ho (X)'' extgreater=
ceil (r+2)/2
ceil$, then $X$ is linearly normal.
- The value of $
ho (X)''$ is strictly less than $2 ext{ceil}((r+1)/(n+1))$, except when $n=r$ or $X$ is a rational normal curve.

## 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.