TY - JOUR

T1 - A relative m-cover of a Hermitian surface is a relative hemisystem

AU - Bamberg, John

AU - Lee, Melissa

N1 - Funding Information:
The authors are indebted to Michael Giudici for many discussions on the material in this work. The first author acknowledges the support of the Australian Research Council Future Fellowship FT120100036. The second author acknowledges the support of a Hackett Postgraduate Scholarship. The authors also thank Daniel Horsley for his mention of an elementary result during his talk at Combinatorics 2016, which was instrumental in the proof of the main result of this paper.
Publisher Copyright:
© 2017, Springer Science+Business Media New York.

PY - 2017/6/1

Y1 - 2017/6/1

N2 - An m-cover of the Hermitian surface H (3 , q2) of PG (3 , q2) is a set S of lines of H (3 , q2) such that every point of H (3 , q2) lies on exactly m lines of S, and 0 < m< q+ 1. Segre (Annali di Matematica Pura ed Applicata Serie Quarta 70:1–201, 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 (J Comb Theory Ser A 118(2):502–509, 2011) introduced the notion of a relative hemisystem of a generalised quadrangle Γ with respect to a subquadrangle Γ′: a set of lines R of Γ disjoint from Γ′ such that every point P of Γ\ Γ′ has half of its lines (disjoint from Γ′) lying in R. In this paper, we provide an analogue of Segre’s result by introducing relative m-covers of generalised quadrangles of order (q2, q) with respect to a subquadrangle and proving that m must be q / 2 when the subquadrangle is doubly subtended. In particular, a relative m-cover of H (3 , q2) with respect to a symplectic subgeometry W (3 , q) is a relative hemisystem.

AB - An m-cover of the Hermitian surface H (3 , q2) of PG (3 , q2) is a set S of lines of H (3 , q2) such that every point of H (3 , q2) lies on exactly m lines of S, and 0 < m< q+ 1. Segre (Annali di Matematica Pura ed Applicata Serie Quarta 70:1–201, 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 (J Comb Theory Ser A 118(2):502–509, 2011) introduced the notion of a relative hemisystem of a generalised quadrangle Γ with respect to a subquadrangle Γ′: a set of lines R of Γ disjoint from Γ′ such that every point P of Γ\ Γ′ has half of its lines (disjoint from Γ′) lying in R. In this paper, we provide an analogue of Segre’s result by introducing relative m-covers of generalised quadrangles of order (q2, q) with respect to a subquadrangle and proving that m must be q / 2 when the subquadrangle is doubly subtended. In particular, a relative m-cover of H (3 , q2) with respect to a symplectic subgeometry W (3 , q) is a relative hemisystem.

KW - Generalised quadrangle

KW - Hermitian surface

KW - Relative hemisystem

UR - http://www.scopus.com/inward/record.url?scp=85010736895&partnerID=8YFLogxK

U2 - 10.1007/s10801-017-0739-5

DO - 10.1007/s10801-017-0739-5

M3 - Article

AN - SCOPUS:85010736895

SN - 0925-9899

VL - 45

SP - 1217

EP - 1228

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

IS - 4

ER -