A Pascal's theorem for rational normal curves

Abstract

Pascal's theorem gives a synthetic geometric condition for six points a , … , f in P 2 to lie on a conic. Namely, that the intersection points a b ¯ ∩ d e ¯ , a f ¯ ∩ d c ¯ , e f ¯ ∩ b c ¯ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for d + 4 points in P d 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 P d 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.