Ruled quartic surfaces, models and classification

TL;DR

This paper revisits the classical classification of ruled quartic surfaces, correcting and completing historical results, and explores their models and related algebraic curves with modern insights.

Contribution

It provides a corrected, modern classification of ruled quartic surfaces and links classical results to contemporary algebraic geometry concepts.

Findings

Corrected classical classification of ruled quartic surfaces

Established a modern proof of Rohn's result on bi-degree (2,2) curves

Connected string models to algebraic curve properties

Abstract

New historical aspects of the classification, by Cayley and Cremona, of ruled quartic surfaces and the relation to string models and plaster models are presented. In a `modern' treatment of the classification of ruled quartic surfaces the classical one is corrected and completed. A conceptual proof is presented of a result of Rohn concerning curves in $\mathbb{P}^1\times \mathbb{P}^1$ of bi-degree $(2,2)$. The string models of Series XIII (of some ruled quartic surfaces) are based on Rohn's result.