A Pascal's theorem for rational normal curves

Pascal's theorem gives a synthetic geometric condition for six points a, horizontal ellipsis ,f in P2 to lie on a conic. Namely, that the intersection points ab over bar boolean AND de over bar , af over bar boolean AND dc over bar , ef over bar boolean AND bc over bar are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for d+4 points in Pd to lie on a degree d 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 d+4-ordered points in Pd 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.