Covers of generalized quadrangles, 2. Kantor-Knuth covers and embedded ovoids

TL;DR

This paper advances the understanding of covers and decomposition laws in generalized quadrangles, especially Kantor-Knuth types, and explores their embedded ovoids and related embeddings.

Contribution

It introduces a higher decomposition law for Kantor-Knuth quadrangles and links ovoid sets to embeddings, addressing previously open questions.

Findings

Established a new decomposition law for Kantor-Knuth quadrangles

Connected ovoid sets to embeddings of parabolic quadrangles

Provided answers to open questions on embeddings and ovoids

Abstract

In this paper, which is a sequel to \cite{part1}, we proceed with our study of covers and decomposition laws for geometries related to generalized quadrangles. In particular, we obtain a higher decomposition law for all Kantor-Knuth generalized quadrangles which generalizes one of the main results in \cite{part1}. In a second part of the paper, we study the set of all Kantor-Knuth ovoids (with given parameter) in a fixed finite parabolic quadrangle, and relate this set to embeddings of parabolic quadrangles into Kantor-Knuth quadrangles. This point of view gives rise to an answer of a question posed in \cite{JATSEP}.