all points of $V$.

Each hyperplane in $\C^{\m(d,r)}$ is given as the set of points having inner product zero with some $w \in \C^{\m(d,r)}$. If we take the corresponding polynomial $f_w \in \C[x_1,\ldots,x_d]$ we get that it vanishes on $V$ iff $\phi_{d,r}(V)$ is contained in the hyperplane defined by $w$.

\[cla-veronese-line\] Suppose the $r+2$ points $v_1,\ldots,v_{r+2} \in \C^d$ are collinear and let $\phi = \phi_{d,r} : \C^d \mapsto \C^{\m(d,r)}$. Then, the points $\phi(v_1),\ldots,\phi(v_{r+2})$ are linearly dependent. Moreover, every $r+1$ of the points $\phi(v_1),\ldots,\phi(v_{r+2})$ are linearly independent.

Denote $u_i = \phi(v_i)$ for $i=1\ldots r+2$. To show that the $u_i$’s are linearly dependent it is enough to show that, for any $w \in \C^{\m(d,r)}$, if all the $r+1$ inner products ${\langle w,u_1 \rangle}, \ldots, {\langle w,u_{r+1} \rangle}$ are zero, then the inner product ${\langle w,u_{r+2} \rangle}$ must also be zero. Suppose this is the case, and let $f_w \in \C[x_1,\ldots,x_d]$ be the polynomial of degree at most $r$ associated with the point $w$ so that ${\langle w,u_i \rangle} = f_w(v_i)$ for all $1 \leq i \leq r+1$. Since the points $v_1,\ldots,v_{r+2}$ are on a single line $L \subset \C^d$, and since the polynomial $f_w$ vanishes on $r+1$ of them, we have that 