Variations on Pascal's Theorem

2024/11/12 by Ciro Ciliberto, Ciliberto, Ciro, Rick Miranda +1
Mathematics · #14N05 #14N15 #14N20 #14N25 #14Q15 #Algebraic Geometry (math.AG) #FOS: Mathematics #History and Theory of Mathematics #Primary 14H45 #Secondary 14A25

paper · pdf · doi:10.48550/arxiv.2411.07638

openalex publication_date 2024/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we present a variety of statements that are in the spirit of the famous theorem of Pascal, often referred to as the Mystic Hexagon. We give explicit equations describing the conditions for d+4 points to lie on rational normal curves. A collection of problems of Pascal type are considered for quadric surfaces in \bbP³. Finally we reprove, using computer algebra methods, a remarkable theorem of Richmond, Segre, and Brown, for quadrics in \bbP⁴ containing five general lines.

Related

• Illumination of Pascal's Hexagrammum and Octagrammum Mysticum
• An Invariant of Algebraic Curves from the Pascal Theorem
• A hyperbolic proof of Pascal's Theorem
• Pascal's Hexagon Theorem implies a Butterfly Theorem in the Complex Projective Plane
• Simplices osculating rational normal curves