Flocks of Cones: Herds and Herd Spaces

TL;DR

This paper aims to establish a foundational theory for flocks of arbitrary cones in projective 3-space, extending the understanding of quadratic cone flocks and their connections to various geometric structures.

Contribution

It introduces the first comprehensive framework for flocks of arbitrary cones in PG(3,q), expanding the scope beyond quadratic cones and exploring their geometric interrelations.

Findings

Connections between quadratic cone flocks and generalized quadrangles

Extension of flocks theory to non-quadratic cones

Potential applications to spreads and hyperovals

Abstract

This is the first in a series of articles devoted to providing a foundation for a theory of flocks of arbitrary cones in PG(3,q). The desire to have such a theory stems from a need to better understand the very significant and applicable special case of flocks of quadratic cones in PG(3,q). Flocks of quadratic cones have connections with several other geometrical objects, including certain types of generalized quadrangles, spreads, translation planes, hyperovals (in even characteristic), ovoids, inversive planes and quasi-fibrations of hyperbolic quadrics. This rich collection of interconnections is the basis for the strong interest in such flocks. Recent work has indicated that some of these connections can be extended to non-quadratic cones, so there is an additional desire to have such a theory.