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

TL;DR

This paper explores the geometric properties of Cayley's ruled cubic surface, showing that Betten-Walker spreads arise from osculating tangents and examining their algebraic nature.

Contribution

It demonstrates that Betten-Walker spreads can be constructed from osculating tangents of Cayley's surface and characterizes their algebraic properties.

Findings

Betten-Walker spreads are linked to osculating tangents of Cayley's surface.

Infinite Betten-Walker spreads are not algebraic sets of lines without modification.

Adding one pencil of lines makes these spreads algebraic.

Abstract

We establish that, over certain ground fields, the set of osculating tangents of Cayley's ruled cubic surface gives rise to a (maximal partial) spread which is also a dual (maximal partial) spread. It is precisely the Betten-Walker spreads that allow for this construction. Every infinite Betten-Walker spread is not an algebraic set of lines, but it turns into such a set by adding just one pencil of lines.