The pseudo-hyperplanes and homogeneous pseudo-embeddings of the generalized quadrangles of order (3, t)

Abstract

In the paper “as reported by De Bruyn (Adv Geom, to appear)”, we introduced the notions of pseudo-hyperplane and pseudo-embedding of a point-line geometry and proved that every generalized quadrangle of order (s, t), 2 ≤ s < ∞, has faithful pseudo-embeddings. The present paper focuses on generalized quadrangles of order (3, t). Using the computer algebra system GAP and invoking some theoretical relationships between pseudo-hyperplanes and pseudo-embeddings obtained in “De Bruyn (Adv Geom, to appear)”, we are able to give a complete classification of all pseudo-hyperplanes of $${\mathcal{Q}}$$ . We hereby find several new examples of tight sets of generalized quadrangles, as well as a complete classification of all 2-ovoids of $${\mathcal{Q}}$$ . We use the classification of the pseudo-hyperplanes of $${\mathcal{Q}}$$ to obtain a list of all homogeneous pseudo-embeddings of $${\mathcal{Q}}$$ .