On the Classification of Cubic Surfaces

✍ Scribed by Bruce, J. W.; Wall, C. T. C.

• Book ID: 111955046
• Publisher: Oxford University Press
• Year: 1979
• Tongue: English
• Weight: 721 KB
• Volume: s2-19
• Category: Article
• ISSN: 0024-6107
• DOI: 10.1112/jlms/s2-19.2.245

✦ Synopsis

Since the classification of (complex, projective) cubic surfaces by their singularities was given by Schlafli [5] over a century ago, the need for a further account may be questioned. We found, however, that the geometry was not particularly simply expounded there or in Cayley [3], and also that the classification is made more transparent by the introduction of terminology and techniques from modern singularity theory. Indeed, the classification is equivalent to one given by Looijenga [4], as we will show at the end, but his paper also fails to make the elementary geometry explicit. This geometry proves to be useful in a detailed study of the partition of the space of cubic surfaces furnished by our classification. Furthermore the classification is very closely related to the universal unfolding of the £ 6 singularity and is of some interest from this viewpoint.
Some singularity theory
We shall use the definitions and results of the papers of Arnol'd [1], . We recall that the " simple " singularities of hypersurfaces have (local) normal forms as follows:A n z ^+ Z z , -2 , ( n ^l ) . 2 D n z i n -1 +z l z 2 2 + y Zz i 2 , (n>4).