Isometries and Collineations of the Cayley Surface

TL;DR

This paper characterizes all symmetries and defining equations of Cayley's ruled cubic surface over various fields, revealing special cases for small fields and a geometric structure of points with a distance function for larger fields.

Contribution

It determines all collineations fixing Cayley's surface and all cubic forms defining it, highlighting exceptional cases for small fields and establishing a non-symmetric distance structure for larger fields.

Findings

All collineations fixing the surface are classified.

Explicit forms of defining cubic equations are provided.

A non-symmetric distance function on the surface's points is constructed.

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 $|K|\geq 4$ 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.