In this paper, we construct an infinite family of hemisystems of the Hermitian surface H(3, q 2 ). In particular, we show that for every odd prime power q congruent to 3 modulo 4, there exists a hemisystem of H(3, q 2 ) admitting C (q 3 +1)/4 : C 3 .2010 Mathematics Subject Classification. 05B25 (primary), 05E30, 51E12 (secondary).
We generalise the work of Segre (1965), Cameron -Goethals -Seidel (1978), and by showing that nontrivial m-ovoids of the dual polar spaces DQ(2d, q), DW(2d − 1, q) and DH(2d − 1, q 2 ) (d 3) are hemisystems. We also provide a more general result that holds for regular near polygons.2010 Mathematics Subject Classification. 05B25, 51E12, 51E20.
Abstract. An m-cover of the Hermitian surface H(3, q 2 ) of PG(3, q 2 ) is a set S of lines of H(3, q 2 ) such that every point of H(3, q 2 ) lies on exactly m lines of S, and 0 < m < q + 1. Segre (1965) proved that if q is odd, then m = (q + 1)/2, and called such a set S of lines a hemisystem. Penttila and Williford (2011) introduced the notion of a relative hemisystem: a set of lines R of H(3, q 2 ), q even, disjoint from a symplectic subgeometry W(3, q) such that every point of H(3, q 2 ) \ W(3, q) lies on exactly q/2 elements of R. In this paper, we provide an analogue of Segre's result by introducing relative m-covers of H(3, q 2 ) with respect to a symplectic subgeometry and proving that m must necessarily be q/2.
Let V be a vector space of dimension d over Fq, a finite field of q elements, and let G ≤ GL(V ) ∼ = GL d (q) be a linear group. A base of G is a set of vectors whose pointwise stabiliser in G is trivial. We prove that if G is a quasisimple group (i.e. G is perfect and G/Z(G) is simple) acting irreducibly on V , then excluding two natural families, G has a base of size at most 6. The two families consist of alternating groups Altm acting on the natural module of dimension d = m − 1 or m − 2, and classical groups with natural module of dimension d over subfields of Fq.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.