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.

Caminata, A., Schaffler, L. (2021). A Pascal's theorem for rational normal curves. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 53(5), 1470-1485 [10.1112/blms.12511].

A Pascal's theorem for rational normal curves

Caminata, A;Schaffler, L
2021-01-01

Abstract

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.
Caminata, A., Schaffler, L. (2021). A Pascal's theorem for rational normal curves. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 53(5), 1470-1485 [10.1112/blms.12511].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/423968
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