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Unformatted text preview: LUIGI CREMONA AND CUBIC SURFACES IGOR V. DOLGACHEV Abstract. We discuss the contribution of Luigi Cremona to the early development of the theory of cubic surfaces. 1. A brief history In 1911 Archibald Henderson wrote in his book [Hen] “While it is doubteless true that the classification of cubic surfaces is complete, the number of papers dealing with these surfaces which continue to appear from year to year furnish abundant proof of the fact that they still possess much the same fascination as they did in the days of their discovery of the twenty-seven lines upon the cubic surface.” It is amazing that a similar statement can be repeated almost a hundred years later. Searching in MathSciNet for “cubic surfaces” and their close cousins “Del Pezzo surfaces” reveals 69 and 80 papers published in recent 10 years. Here are some of the highlights in the history of classical research on cubic surfaces before the work of Cremona. A good source is Pascal’s Repertorium [Pas]. 1849 : Arthur Cayley communicates to George Salmon that a general cubic surface contains a finite number of lines. Salmon proves that the number of lines must be equal to 27. Salmon’s proof is presented in Cayley’s paper [Cay]. In the same paper Cayley shows that a general cubic surface admits 45 tritangent planes, i.e. planes planes which intersect the surface along the union of three lines. He gives a certain 4-parameter family of cubic surfaces for which the equations of tritangent planes can be explicitly found and their coefficients are rational functions in parameters. In a paper published in the same year and in the same journal [Sal1] Salmon proves that not only a general but any nonsingular surface contains exactly 27 lines. He also finds the number of lines on 11 different types of singular surfaces. The discovery of 27 lines on a general cubic surface can be considered as the first non-trivial result on algebraic surfaces of order higher than 2. In fact, it can be considered as the beginning of modern algebraic geometry. Research is partially supported by NSF grant DMS-0245203. 1 2 IGOR V. DOLGACHEV 1851 : John Sylvester claims without proof that a general cubic surface can be written uniquely as a sum of 5 cubes of linear forms [Syl]: F 3 = L 3 1 + L 3 2 + L 3 3 + L 3 4 + L 3 5 . This was proven ten years later by Alfred Clebsch [Cle3]. The union of planes L i = 0 will be known as the Sylvester pentahedron of the cubic surface. 1854 : Ludwig Schl¨ afli finds about the 27 lines from correspondence with Jacob Steiner [Graf]. In his letters he communicates to Steiner some of his results on cubic surfaces which were published later in 1858 [Schl1]. For example he shows that a general cubic surface has 36 double-sixes of lines. A double-six is a pair of sets of 6 skew lines ( sixes ) such that each line from one set is skew to a unique line from the other set making a bijection between the two sets. He introduces a new notation for the 27 lines ( a i , b i , c...
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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