Characterising elliptic and hyperbolic hyperplanes of the parabolic quadric \Q(2n,q)

TL;DR

This paper characterizes elliptic and hyperbolic hyperplanes of the parabolic quadric Q(2n,q) for even q, based on element counts through points and codimension 2 spaces, extending previous characterizations.

Contribution

It provides a natural characterization of these hyperplanes for even q, generalizing earlier specific cases for elliptic and hyperbolic solids.

Findings

Characterization based on element counts through points.

Extension of previous characterizations to higher dimensions.

Applicable for even q in parabolic quadrics.

Abstract

We provide a natural characterisation for the sets of elliptic and hyperbolic hyperplanes of the parabolic quadric Q(2n,q) when q is even. This characterisation is based on the number of elements of these sets through points and codimension 2 spaces and generalises [S. Barwick, A. Hui, and W-A. Jackson. Characterising elliptic solids of Q(4,q), q even. Discrete Math., 343 (6) (2020), 111857] and [S. Barwick, A. Hui, W-A. Jackson, and J. Schillewaert. Characterising hyperbolic solids of Q(4,q), q even. Des. Codes Cryptogr., 88 (1) (2020), 33--39.].