A Pascal's theorem for rational normal curves

Pascal's theorem gives a synthetic geometric condition for six points (Formula presented.) in (Formula presented.) to lie on a conic. Namely, that the intersection points (Formula presented.), (Formula presented.), (Formula presented.) are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for (Formula presented.) points in (Formula presented.) to lie on a degree (Formula presented.) rational normal curve? In this paper we find many of these conditions by writing in the Grassmann–Cayley algebra the defining equations of the parameter space of (Formula presented.) -ordered points in (Formula presented.) that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic.