For $k>2, d>2$, is it possible for the Veronese variety $v\_k(\mathbb P^{d-1})$ to contain $k+1$ points that are not linearly independent?


For $d = 2$, an answer in the negative is given by the Vandermonde determinant. Is a similar statement true in general?



Can the Veronese variety of degree k have k+1 linearly dependent points?
Distinct points $x\_1,\dots,x\_{k+1}$ are always sent to linearly independent points in $v\_k(\mathbb P^{d-1})$. Let $f\_i$ be a linear form that vanishes at $x\_i$ but not at $x\_{k+1}$. Let $f$ be the product of all the $f\_i$. Since $f$ is a form of degree $k$, it gives a linear form on the $k$th symmetric power which vanishes at the linear span of all $v\_k(x\_i)$ but not $v\_k(x\_{k+1})$.


