$\def\PP{\mathbb{P}}$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 $V$?



Equations for points to lie on a rational normal curve
Let me try the $\mathbb P^3$ case, and only look for a non-degenerate rational normal curve. Apply a linear map to move the first four of your points to be the standard four points ($[1,0,0,0]$,...) (if the four are coplanar, they're not on an rnc). Then apply the standard Cremona involution inverting each coordinate. Do the remaining points become collinear? If yes, they were on a rational normal curve (pre-Cremona). If no, they weren't. (For example, with six points, the remaining two are always collinear; not so with seven!)


I think this works the same in any dimension, at least generically. If we're lucky, the closure of this relation (to include linearly degenerate configurations) will be the right equations to check if the points are on a degenerate rnc.


