The known maximal partial ovoids of size $q^2-1$ of Q(4,q)

TL;DR

This paper characterizes maximal partial ovoids of size q^2-1 in the parabolic quadric Q(4,q) using group actions and spread sets, providing explicit descriptions and connections to root systems.

Contribution

It offers a new group-theoretic and geometric framework for understanding maximal partial ovoids of size q^2-1 in Q(4,q), including explicit classifications.

Findings

Maximal partial ovoids correspond to sharply transitive subsets of SL(2,q).

Explicit descriptions of known examples are provided.

Connections to root systems are established.

Abstract

We present a description of maximal partial ovoids of size $q^2-1$ of the parabolic quadric $\q(4,q)$ as sharply transitive subsets of $\SL(2,q)$ and show their connection with spread sets. This representation leads to an elegant explicit description of all known examples. We also give an alternative representation of these examples which is related to root systems.