Translation generalized quadrangles of order \((s,s ^{2})\), \(s\) even, and eggs
An egg \(O(n,2n,q)\) of \(PG(4n-1,q)\) is a set of \(q^{2n}+1\) \((n-1)\)-dimensional subspaces \(PG^{(i)}(n-1,q)\), \(i=0,\ldots,q^{2n}\), any three of which generate a \(PG(3n-1,q)\) and such that every element \(PG^{(i)}(n-1,q)\) of \(O(n,2n,q)\) is contained in a \(PG^{(i)}(3n-1,q)\) having no point in common with any element \(PG^{(j)}(n-1,q)\), \(j \neq i\). Similarly, a set \(O(n,n,q)\) of \(PG(3n-1,q)\) is a set of \(q^{n}+1\) \((n-1)\)-dimensional subspaces \(PG^{(i)}(n-1,q)\), \(i=0,\ldots,q^{n}\), any three of which generate a \(PG(3n-1,q)\) and such that every element \(PG^{(i)}(n-1,q)\) of \(O(n,n,q)\) is contained in a \(PG^{(i)}(2n-1,q)\) having no point in common with any element \(PG^{(j)}(n-1,q)\), \(j \neq i\). The easiest examples of an egg and of an \(O(n,n,q)\) are obtained by considering an ovoid of \(PG(3,q^n)\) and by considering an oval of \(PG(2,q^n)\). When this latter ovoid of \(PG(3,q^n)\) is the elliptic quadric, we call the egg classical and when this latter oval of \(PG(2,q^n)\) is a conic, we call the set \(O(n,n,q)\) a pseudo-conic. An egg \(O(n,2n,q)\) is good at its element \(PG^{(i)}(n-1,q)\) if any \(PG(3n-1,q)\) containing \(PG^{(i)}(n-1,q)\) and at least two other elements of \(O(n,2n,q)\) contains exactly \(q^n+1\) elements of \(O(n,2n,q)\). The author presents two characterization results on classical eggs. He first of all proves that the egg \(O(n,2n,q)\) of \(PG(4n-1,q)\), \(q\) even, is classical if and only if it is good at some element and contains at least one pseudo-conic. And secondly, he proves that the egg \(O(n,2n,q)\) of \(PG(4n-1,q)\), \(q\) even, is classical if and only if it contains at least two intersecting pseudo-conics. As an application, three characterization results of the generalized quadrangle \(Q^-(5,q)\), \(q\) even, are given.
scientific article; zbMATH DE number 1888399