Lax embeddings of generalized quadrangles in finite projective spaces (Q2766396)

A generalized quadrangle \({\mathcal S}=(P,B,I)\) is laxly embedded in the projective space \(PG(d,q)\) with \(d\geq 2\), if (i) \(P\) is a point set of \(PG(d,q)\) which generates \(PG(d,q)\); (ii) each line \(L\) of \(\mathcal S\) is a subset of a line \(L'\) of \(PG(d,q)\), and distinct lines of \(\mathcal S\) define distinct lines of \(PG(d,q)\).NEWLINENEWLINENEWLINETheorem 1.3. If the generalized quadrangle \({\mathcal S}\) of order \((s,t)\) with \(s>1\) is laxly embedded in the projective space \(PG(d,q)\), then \(d\leq 5\). Furthermore (i) if \(d=5\), then \({\mathcal S}\simeq Q(5,s)\); (ii) if \(d=4\), then \(s\leq t\) and (a) if \(s=t\), then \({\mathcal S}\simeq Q(4,s)\); (b) if \(t=s+2\), then \(s=2\) and \({\mathcal S}\simeq Q(5,2)\); (c) if \(t^2=s^3\), then \({\mathcal S}\simeq H(4,s)\); (iii) if \(d=3\) and \(s=t^2\), then \({\mathcal S}\simeq H(3,s)\).NEWLINENEWLINENEWLINEIn two appendices the followingresults are proved.NEWLINENEWLINENEWLINETheorem A.1. All the points of a generalized quadrangle \({\mathcal S}\) of order \((s,s-2)\), with \(s\geq 4\), are regular if and only if \(s=4\) and \({\mathcal S}\simeq H(3,4)\).NEWLINENEWLINENEWLINETheorem B.1. A generalized quadrangle \({\mathcal S}\) of order \((s^2,s^3)\) is isomorphic to \(H(4,s^2)\) if and only if any two nonconcurrent lines are contained in a proper subquadrangle of order \((s^2,t)\), with \(t\neq 1\).