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Ruled quartic surfaces, models and classification. (English) Zbl 1226.14048

Summary: New historical aspects of the classification, by A. Cayley [Trans. of Lond. CLIX, 111–126 (1869; JFM 02.0598.01)] and L. Cremona [Rend. d. Ist. Lomb. (2) I. 420 (1868; JFM 01.0130.02)], 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. The string models of Series XIII of some ruled quartic surfaces (manufactured by L. Brill and by M. Schilling [Katalog mathematischer Modelle für den höheren mathematischen Unterricht. Leipzig: Martin Schilling (1911; JFM 42.1037.04)] are based on a result of Rohn concerning curves in \(\mathbb{P}^{1}\times \mathbb{P}^{1}\) of bi-degree \((2, 2)\). This is given here a conceptional proof.

MSC:

14J26 Rational and ruled surfaces
14-03 History of algebraic geometry
01A60 History of mathematics in the 20th century
14M15 Grassmannians, Schubert varieties, flag manifolds
14N25 Varieties of low degree
32S25 Complex surface and hypersurface singularities
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References:

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