TITLE: Can the Veronese variety of degree k have k+1 linearly dependent points?
QUESTION [5 upvotes]: 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?

REPLY [9 votes]: 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 $v_k(x_{k+1})$.