$\def\PP{\mathbb{P}}$Let $z_1$, $z_2$, ..., $z_n$ be points in $\PP^{k-1}$. I am interested in equations for when the $z_i$ lie on a rational normal curve (or degeneration thereof.)

Specifically, let $s: \PP^{k-1} \to \PP^{\binom{k+1}{2} - 1}$ be the $2$-uple Veronese. If the $z_i$ lie on a rational normal curve, then the $n \times \binom{k+1}{2}$ matrix $(s(z_1), s(z_2), \ldots, s(z_n))$ has rank $2k-1$.

Let $X$ be the subscheme of $(\PP^{k-1})^n$ where $\mathrm{rank}(s(z_1), s(z_2), \ldots, s(z_n)) \leq 2k-1$. Let $V$ be the subscheme of $(\PP^{k-1})^n$ obtained by taking the closure of those points that lie on a degree $k-1$ rational curve. So $V \subset X$. If $k \leq 2$, this is trivial. If $k=3$, this says that $n$ points lie on a conic if and only if the $6 \times n$ matrix $(s(z_1), s(z_2), \ldots, s(z_n))$ has a right kernel -- again, this is obvious. So the first hard situation is $k=4$.

I would love it if $V=X$. That seems too good to be true, once $k \geq 4$, although I don't actually have a counter-example. Here are weaker things that would make me happy:

Is $V$ an irreducible component of $X$?

Is $X$ reduced at a generic point of $V$?

Is there some explicit description of the other components of $V$?