# A Pascal's Theorem for rational normal curves

## TL;DR

This paper generalizes Pascal's Theorem to higher dimensions by deriving coordinate-free conditions for points to lie on rational normal curves using Grassmann-Cayley algebra, with applications to seven points on a twisted cubic.

## Contribution

It provides new algebraic equations characterizing when points in projective space lie on rational normal curves, extending classical geometric theorems to higher dimensions.

## Key findings

- Derived equations for points on rational normal curves in projective space.
- Extended Pascal's Theorem to higher-dimensional rational normal curves.
- Applied results to seven points on a twisted cubic.

## Abstract

Pascal's Theorem gives a synthetic geometric condition for six points $a,\ldots,f$ in $\mathbb{P}^2$ to lie on a conic. Namely, that the intersection points $\overline{ab}\cap\overline{de}$, $\overline{af}\cap\overline{dc}$, $\overline{ef}\cap\overline{bc}$ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for $d+4$ points in $\mathbb{P}^d$ 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 $\mathbb{P}^d$ 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.