Finite generalized quadrangles as the union of few large subquadrangles (Q1808827)

Define a large subquadrangle of a generalized quadrangle of order \((s,t)\) as a subquadrangle of order \((s,t')\) with \(t'<t\). In this paper the author gives a fairly general answer to the question: how many large subquadrangles do we need to cover a (point set) generalized quadrangle? Theorem 2. Let \(\Gamma\) be a GQ of order \((s,t)\) with \(s>1\), \(t>1\). Then \(\Gamma\) cannot be the union fewer than \(s+1\) large subquadrangles. Also, if \(\Gamma\) is the union \(\mathcal S\) of \(s+1\) subquadrangles with \(s>2\), these subquadrangles all have the same order \((s,t')\), and one of the following holds: (i) the point set \(\Gamma\) is the disjoint union of the point sets of the members of \(\mathcal S\) and \(t'=(t-1)/(s+1)\); (ii) there exists a large subquadrangle \(\Gamma^*\) of order \((s,1)\) such that every two members of \(\mathcal S\) meet precisely in \(\Gamma^*\). Every member of \(\mathcal S\) has order \((s,s)\), and \(t=s^2\); (iii) \((t',s,t)=(2,4,8)\), every two members of \(\mathcal S\) meet in the nine points of an ovoid in both members, there are exactly 30 points of \(\Gamma\), which lie in at least two members of \(\mathcal S\) and every such point lies in exactly 3 members, every member contains exactly 18 points which lie in three members of \(\mathcal S\) and no line is contained in at least two members of \(\mathcal S\); (iv) \((t',s,t)=(1,3,3)\) and there are exactly two nonisomorphic examples, one with no line of \(\Gamma\) in at least two members of \(\mathcal S\), and the other with two unique concurrent lines contained in 3 members of \(\mathcal S\); (v) \((t',s,t)=(10,15,160)\) and there exists a line \(L\) of \(\Gamma\) such that every two members of \(\mathcal S\) meet precisly in \(L\). Theorem 3. Let \(\Gamma\) be a finite polar space of rank \(r\) naturally embedded in \(PG(d,q)\). Suppose that \(\Gamma\) is the union of \(k\leq q+1\) large polar subspaces of rank \(k\), and that \(q>2\) if \(r=2\). Then \(k=q+1\) and either \(r=2\) and one of the cases (iii) or (iv) of Theorem 2 holds, or \(\Gamma\) is an elliptic quadric and there exist \(q+1\) hyperplanes of \(PG(d,q)\) containing a \((d-2)\)-dimensional space \(U\) such that each hyperplane meets \(\Gamma\) precisly in a large polar subspace (which is a parabolic quadric). Also, \(U\) meets \(\Gamma\) in a large polar subspace of rank \(r\) (which is a hyperbolic quadric).