2011/08/16 by Will Traves, Traves, Will
Computer Science · Engineering · Mathematics · #13-03 #13H10 (Secondary) #14H50 (Primary) #14N05 #14N15 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #History and Overview (math.HO) #History and Theory of Mathematics #Mathematics and Applications #Point processes and geometric inequalities #Polynomial and algebraic computation #math.AC #math.AG #math.HO #msc:13-03 #msc:13H10 #msc:14H50 #msc:14N05 #msc:14N15
paper · pdf · doi:10.48550/arxiv.1108.3368
26 pages with 6 figures
arxiv created 2011/08/16 · openalex publication_date 2011/08/16 · arxiv updated 2011/08/18 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Using a new point of view inspired by hyperplane arrangements, we generalize the converse to Pascal's Theorem, sometimes called the Braikenridge-Maclaurin Theorem. In particular, we show that if 2k lines meet a given line, colored green, in k triple points and if we color the remaining lines so that each triple point lies on a red and blue line then the points of intersection of the red and blue lines lying off the green line lie on a unique curve of degree k-1. We also use these ideas to extend a second generalization of the Braikenridge-Maclaurin Theorem, due to Möbius. Finally we use Terracini's Lemma and secant varieties to show that this process constructs a dense set of curves in the space of plane curves of degree d, for degrees d <= 5. The process cannot produce a dense set of curves in higher degrees. The exposition is embellished with several exercises designed to amuse the reader.