A Pascal's Theorem for rational normal curves
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2019
- DOI:
- 10.48550/arXiv.1903.00460
- arXiv:
- arXiv:1903.00460
- Bibcode:
- 2019arXiv190300460C
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14A25;
- 14H50;
- 51N35
- E-Print:
- 17 pages, 1 figure. Final version. To appear in Bulletin of the London Mathematical Society