Isometries and collineations of the Cayley surface

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 nonsymmetric distance function. We describe the corresponding circles, and we establish that each isometry extends to a unique projective collineation of the ambient space.

Counting rational points on the Cayley ruled cubic

“…While (1) is plainly toric the Cayley surface is not toric. Indeed, according to Gmainer and Havlicek [10,Lemma 3.1], the automorphism group of W is a 3-dimensional algebraic group, which contains a 2-dimensional unipotent subgroup. Thus there is no 2-dimensional torus acting faithfully on W .…”

The Betten-Walker spread and Cayley's ruled cubic surface

“…We refer to [1], [3], [4], [8], [20], and [22] for the definition and basic properties of Cayley's ruled cubic surface or, for short, the Cayley surface. It is, to within projective collineations, the point set…”

“…Under the action of G, the points of F fall into three orbits: F \ ω, g ∞ \ {Z}, and {Z}. Except for the case when |K| ≤ 3, the group G yields all projective collineations of F ; see [8,Section 3].…”