Innovations in Incidence Geometry

Isometries and collineations of the Cayley surface

Johannes Gmainer and Hans Havlicek

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Abstract

Let F be Cayley’s ruled cubic surface in a projective three-space over any commutative field K . We determine all collineations fixing F , as a set, and all cubic forms defining F . For both problems the cases | K | = 2 , 3 turn out to be exceptional. On the other hand, if | K | 4 then the set of simple points of F can be endowed with a non-symmetric distance function. We describe the corresponding circles, and we establish that each isometry extends to a unique projective collineation of the ambient space.

Article information

Source
Innov. Incidence Geom., Volume 2, Number 1 (2005), 109-127.

Dates
Received: 23 November 2004
Accepted: 17 March 2005
First available in Project Euclid: 28 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551323268

Digital Object Identifier
doi:10.2140/iig.2005.2.109

Mathematical Reviews number (MathSciNet)
MR2214718

Zentralblatt MATH identifier
1096.51007

Subjects
Primary: 51B15: Laguerre geometries 51N25: Analytic geometry with other transformation groups 51N35: Questions of classical algebraic geometry [See also 14Nxx]

Keywords
Cayley surface non-symmetric distance isometry

Citation

Gmainer, Johannes; Havlicek, Hans. Isometries and collineations of the Cayley surface. Innov. Incidence Geom. 2 (2005), no. 1, 109--127. doi:10.2140/iig.2005.2.109. https://projecteuclid.org/euclid.iig/1551323268


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