Variations on Pascal's Theorem

TL;DR

This paper explores generalizations of Pascal's theorem, providing explicit equations for points on rational normal curves, examining Pascal-type problems on quadric surfaces, and re-proving a theorem about quadrics containing lines using computer algebra.

Contribution

It introduces new equations for points on rational normal curves, extends Pascal-type problems to quadric surfaces, and re-proves a significant theorem with computer algebra methods.

Findings

Explicit equations for points on rational normal curves.

Pascal-type problems on quadric surfaces in projective space.

Reproof of a theorem on quadrics containing lines using computer algebra.

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 $\bbP^3$. Finally we reprove, using computer algebra methods, a remarkable theorem of Richmond, Segre, and Brown, for quadrics in $\bbP^4$ containing five general lines.