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.

I would like to suggest the following paper, where my coauthors and I try to give a partial answer to this question

https://arxiv.org/abs/1711.06286

Roughly speaking, the idea is to use the Gale transform to reduce to the planar situation. In fact, one has that (d+4) points in P^d lie on a rational normal curve iff their Gale duals lie on a conic in P^2.

Uggh, I lose. $7$ generic points in $\mathbb{P}^3$ don't lie on a rational normal curve, but apparently they DO lie on $3$ quadratics. (Because we are looking at the kernel of a $10 \times 7$ matrix.)

That means there is some fun Chasles theorem stuff here: Any quadratic which passes through $7$ of the intersection points of three quadratics should pass through the $8$th.

So my specific suggestion of using quadratics was wrong. I'd be glad to hear any suggestions as to what is right.