TY - JOUR

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

AU - Bamberg, John

AU - Lee, Melissa

PY - 2017/6/1

Y1 - 2017/6/1

N2 - An m-cover of the Hermitian surface (Formula presented.) of (Formula presented.) is a set (Formula presented.) of lines of (Formula presented.) such that every point of (Formula presented.) lies on exactly m lines of (Formula presented.), and (Formula presented.). Segre (Annali di Matematica Pura ed Applicata Serie Quarta 70:1–201, 1965) proved that if q is odd, then (Formula presented.), and called such a set (Formula presented.) 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 (Formula presented.) with respect to a subquadrangle (Formula presented.): a set of lines (Formula presented.) of (Formula presented.) disjoint from (Formula presented.) such that every point P of (Formula presented.) has half of its lines (disjoint from (Formula presented.)) lying in (Formula presented.). In this paper, we provide an analogue of Segre’s result by introducing relative m-covers of generalised quadrangles of order (Formula presented.) 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 (Formula presented.) with respect to a symplectic subgeometry (Formula presented.) is a relative hemisystem.

AB - An m-cover of the Hermitian surface (Formula presented.) of (Formula presented.) is a set (Formula presented.) of lines of (Formula presented.) such that every point of (Formula presented.) lies on exactly m lines of (Formula presented.), and (Formula presented.). Segre (Annali di Matematica Pura ed Applicata Serie Quarta 70:1–201, 1965) proved that if q is odd, then (Formula presented.), and called such a set (Formula presented.) 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 (Formula presented.) with respect to a subquadrangle (Formula presented.): a set of lines (Formula presented.) of (Formula presented.) disjoint from (Formula presented.) such that every point P of (Formula presented.) has half of its lines (disjoint from (Formula presented.)) lying in (Formula presented.). In this paper, we provide an analogue of Segre’s result by introducing relative m-covers of generalised quadrangles of order (Formula presented.) 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 (Formula presented.) with respect to a symplectic subgeometry (Formula presented.) 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

UR - https://arxiv.org/abs/1608.03055

U2 - 10.1007/s10801-017-0739-5

DO - 10.1007/s10801-017-0739-5

M3 - Article

AN - SCOPUS:85010736895

VL - 45

SP - 1217

EP - 1228

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

IS - 4

ER -