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

TL;DR

This paper extends the converse to Pascal's Theorem using hyperplane arrangements and the Cayley-Bacharach Theorem, revealing new geometric relations and constructing dense sets of plane curves up to degree 5.

Contribution

It generalizes the converse to Pascal's Theorem through hyperplane arrangements and secant varieties, providing new geometric insights and constructions.

Findings

The generalized theorem applies to curves of degree up to 5.

The process constructs a dense set of curves in the space of plane curves of degree d <= 5.

The method cannot produce dense sets for degrees higher than 5.

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 M\"obius. 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.