Equations for point configurations to lie on a rational normal curve

The parameter space of $n$ ordered points in projective $d$-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in $(\mathbb{P}^d)^n$. The resulting variety was used to study the birational geometry of the moduli space $\overline{\mathrm{M}}_{0,n}$ of $n$-tuples of points in $\mathbb{P}^1$. In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely $d=2$, we find scheme-theoretic equations revealing a determinantal structure and use this to prove some geometric properties; moreover, determining which subsets of these equations suffice set-theoretically is equivalent to a well-studied combinatorial problem. For twisted cubics, $d=3$, we use the Gale transform to produce equations defining the union of two irreducible components, the compactified configuration space we want and the locus of degenerate point configurations, and we explain the challenges involved in eliminating this extra component. For $d \ge 4$ we conjecture a similar situation and prove partial results in this direction.