Vol. 2, No. 1, 2005

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Isometries and collineations of the Cayley surface

Johannes Gmainer and Hans Havlicek

Vol. 2 (2005), No. 1, 109–127
DOI: 10.2140/iig.2005.2.109
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.

Keywords
Cayley surface, non-symmetric distance, isometry
Mathematical Subject Classification 2000
Primary: 51B15, 51N25, 51N35
Milestones
Received: 23 November 2004
Accepted: 17 March 2005
Authors
Johannes Gmainer
Hans Havlicek