Covers and blocking sets of classical generalized quadrangles (Q5943883)

A cover of a finite generalized quadrangle \(Q\) is a set \({\mathcal C}\) of lines such that each point of \(Q\) is contained in at least one line of \({\mathcal C}\). A spread is a cover whose lines are pairwise disjoint. Note that the quadric \(Q(4,q)\) contains a spread if and only if \(q\) is even. NEWLINENEWLINENEWLINEThe authors prove the following theorems: NEWLINENEWLINENEWLINETheorem 1. Let \({\mathcal C}\) be a cover of \(Q(4, q)\), \(q\) odd. Then \(|{\mathcal C}|> q^2 + 1 + (q-1)/3\). NEWLINENEWLINENEWLINETheorem 2. Let \({\mathcal C}\) be a cover of \(Q(4, q)\), \(q \geq 32\), \(q\) even. If \(|{\mathcal C}|\leq q^2 + 1 + \sqrt{q}\), then \({\mathcal C}\) contains a spread of \(Q(4, q)\). NEWLINENEWLINENEWLINEThe paper also contains the following interesting result on blocking sets in the generalized quadrangle \(U(4, q^2)\): NEWLINENEWLINENEWLINETheorem 3. The generalized quadrangle \(U(4, q^2)\) does not contain minimal blocking sets of size \(q^5+2\) for \(q>2\), of size \(q^5+3\) for \(q>3\) and of size \(q^5+4\) for \(q>4\).