Let $X\subseteq \mathbb{P}^n$ be a quasi-projective variety. I will say every points of $X$ are in general position if any $n+1$ points of $X$ are linearly independent. I know every points of rational normal curve are in general position.
For Veronese map, generalization of rational normal curve, this is not always true. For example $[x^2:x:x:1:1:1]$ are in Veronese surface but $6$ points of them are not linearly independent.
Is there exist any other variety satisfying this condition? Or someone proved rational normal curve is the only quasi-projective variety satisfying this condition?