Bounds on partial ovoids and spreads in classical generalized quadrangles (Q447733)

A finite generalized quadrangle of order \((s,t)\) is a finite point-line incidence geometry such that two distinct lines intersect in at most one point; every line is incident with exactly \(s+1\) points; every point is incident with exactly \(t+1\) lines and given any point \(p\) not on a given line \(L\) there exists a unique line \(M\) containing \(p\) and intersecting \(L\).NEWLINENEWLINE An ovoid \(\mathcal{O}\) of a generalized quadrangle \(\mathbb{Q}\) is a set of points of \(\mathbb{Q}\) such that each line contains a unique point of \(\mathcal{O}\). A partial ovoid is a set of mutually non-collinear points of \(\mathbb{Q}\). A partial ovoid is called maximal if it cannot be extended to a larger partial ovoid.NEWLINENEWLINE Let \(W(q)\) be the generalized quadrangle with set of points \(\mathrm{PG}(3,q)\) and set of lines all the lines totally isotropic with respect to a symplectic polarity. In this work the authors prove the non-existence of maximal partial ovoids of size \(q^3+q^2+q+1\) in the generalized quadrangle \(W(q^3)\), \(q=p^h\), \(h\geq 7\).NEWLINENEWLINE Let \(Q(r,q)\), with \(r=3,4,5\), be the generalized quadrangle with set of points the points of a non-singular quadrics \(Q\) of projective index \(1\) contained in \(\mathrm{PG}(r,q)\) and set of lines the lines of \(Q\). The authors classify partial ovoids \(\mathcal{K}\) of the generalized quadrangle \(Q(4,q)\), \(q=5,7,11\), of size \(q^2-1\) having the property that \((q^2-1)^2\) divides the size of \(\mathrm{Aut}(Q(4,q))_{\mathcal{K}}\).