Ruled Quartic Surfaces, Models and Classification

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 P × P of bi-degree (2, 2). The string models of Series XIII (of some ruled quartic surfaces) are based on Rohn’s result. Motivation and History The collection of string models of ruled quartic surfaces, present at some mathematical institutes (for instance at the department of mathematics in Groningen) is the direct motivation for this paper. This Series XIII, produced by Martin Schilling in 1886, is based upon a paper of K. Rohn [13] containing a classification of ruled quartic surfaces over C and R. Some authors before Rohn (e.g., M. Chasles [4], A. Cayley [3], L. Cremona [5], R. Sturm [17], G. Salmon [14]) and many after his time (e.g., B.C. Wong [23], H. Mohrmann [10], W.Fr. Meyer [9], W.L. Edge [7], O. Bottema [2], T. Urabe [20]) have contributed to this beautiful topic of 19th century geometry. Cremona classified the ruled quartic surfaces in 12 types. He states in [5] that Cayley produced 8 of these without revealing his method. However, Cayley’s third memoir on this subject [3] was written earlier the same year 1868 and contains 10 types. In an addition to this memoir (May 18, 1869), Cayley gives the comparison between his own classification and the one by Cremona and makes it clear what the two types he missed are. The method of Cayley consists of taking three curves in P and to consider the ruled surface S which is the union of the lines meeting all three curves. Using a formula for the degree of S, he now computes possibilities of ruled quartic surfaces. The expression ‘the six coordinates of a line’ in Cayley’s work indicates that the Grassmann variety Gr(2, 4) of the lines in P plays a role. The work of Cayley contains also explicit calculations for reciprocal surfaces (see Date: April 15, 2009. 1 ar X iv :0 90 4. 23 74 v1 [ m at h. A G ] 1 5 A pr 2 00 9 2 IRENE POLO-BLANCO, MARIUS VAN DER PUT AND JAAP TOP