Variations on Pascal’s theorem

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 \({\mathbb {P}}^3\). Finally we re-prove, using computer algebra methods, a remarkable theorem of Richmond, Segre, and Brown for quadrics in \({\mathbb {P}}^4\) containing five general lines.