Generalized quadrangles of order \((s,s^2)\): Recent results (Q1808826)

This paper contains recent results on generalized quadrangles (GQ) of order \((s,s^2)\). In particular, it contains theorems on 3-regularity, the Axiom of Veblen, subquadrangles, Property \((G)\), translation generalized quadrangles and Veronese Surfaces. For example Corollary 3.2 (Thas and Van Maldeghem). Let \({\mathcal S}\) be a GQ of order \((s,t)\) with \(s\neq t\), \(s>1\), \(t>1\). \((i)\) If \(s\) is odd, then \({\mathcal S}\) is isomorphic to the classical GQ \(Q(5,s)\) if and only if it has a coregular point \(x\) and if for each line \(L\) incident with \(x\) the corresponding dual net \({\mathcal N}^*_L\) satisfies the Axiom of Veblen. \((ii)\) If \(s\) is even, then \({\mathcal S}\) is isomorphic to the classical GQ \(Q(5,s)\) if and only if all its lines are regular and if for at least one point \(x\) and all lines \(L\) incident with \(x\) the dual net \({\mathcal N}^*_L\) satisfy the Axiom of Veblen. Let \({\mathcal S}\) be a GQ of order \((s,t)\) with \(s>1\), \(t>1\). A collineation \(\theta\) of \({\mathcal S}\) is an elation about the point \(p\) if \(\theta=id\) or if \(\theta\) fixes all lines incident with \(p\) and fixes no point of \(P-p^\bot\). If there is an abelian group \(H\) of elations about \(p\) acting regularly on \(P-p^\bot\), we say that \(({\mathcal S}^{(p)},H)\) is a translation generalized quadrangle (TGQ) with base point \(p\). For any TGQ \({\mathcal S}^{(p)}\) the point \(p\) is coregular, if \(s\neq t\) then \(s=q^a,t=q^{a+1}\), with \(q\) a prime power and \(a\) an odd integer, further if \(s\) is also even, then \(a=1\). Theorem 4.7 (Thas and Van Maldeghem). Let \({\mathcal S}^{(p)}\) be a TGQ of order \((s,s^2)\), \(s\) odd and \(s>1\). If the dual net \({\mathcal N}^*_L\) defined by some regular line \(L\), with \(p\in L\), satisfies the Axiom of Veblen, then \({\mathcal S}^{(p)}\) contains at least \(s^3+s^2\) classical subquadrangles \(Q(4,s)\).