Generalizing the Converse to Pascal's Theorem via Hyperplane Arrangements and the Cayley-Bacharach Theorem

Abstract

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.