Covers and blocking sets of classical generalized quadrangles

Abstract

This article discusses two problems on classical generalized quadrangles. It is known that the generalized quadrangle Q(4, q) arising from the parabolic quadric in PG(4, q) has a spread if and only if q is even. Hence, for q odd, the problem arises of the cardinality of the smallest set of lines of Q(4, q) covering all points of Q(4, q). We show in this paper that this set of lines must contain more than q 2+1+( q−1)/3 lines. We also show that Q(4, q), q even, does not contain minimal covers of sizes q 2+1+ r when q⩾32 and 0<r⩽ q . To obtain this latter result, we generalize a result on minimal covers of lines in PG(3, q) to minimal covers of lines of the classical generalized quadrangles. This result is then also used to study minimal blocking sets of the non-singular generalized quadrangle U(4, q 2) arising from the Hermitian variety in PG(4, q 2). It is known that U(4, q 2) does not have an ovoid. Here, we show that it also does not contain minimal blocking sets of sizes q 5+2, q 5+3 and q 5+4 except maybe for small values of q.