Abstract
An m-cover of the Hermitian surface \(\mathrm {H}(3,q^2)\) of \(\mathrm {PG}(3,q^2)\) is a set \(\mathcal {S}\) of lines of \(\mathrm {H}(3,q^2)\) such that every point of \(\mathrm {H}(3,q^2)\) lies on exactly m lines of \(\mathcal {S}\), 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 \(\mathcal {S}\) 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 \(\varGamma \) with respect to a subquadrangle \(\varGamma '\): a set of lines \(\mathcal {R}\) of \(\varGamma \) disjoint from \(\varGamma '\) such that every point P of \(\varGamma \setminus \varGamma '\) has half of its lines (disjoint from \(\varGamma '\)) lying in \(\mathcal {R}\). In this paper, we provide an analogue of Segre’s result by introducing relative m-covers of generalised quadrangles of order \((q^2,q)\) 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 \(\mathrm {H}(3,q^2)\) with respect to a symplectic subgeometry \(\mathrm {W}(3,q)\) is a relative hemisystem.
Similar content being viewed by others
References
Brown, M.R.: Generalized Quadrangles and Associated Structures. Ph.D. Thesis, University of Adelaide (1997)
Cameron, P.J.: Partial quadrangles. Q. J. Math. Oxf. Ser. 2(26), 61–73 (1975)
Cameron, P.J., Goethals, J.-M., Seidel, J.J.: Strongly regular graphs having strongly regular subconstituents. J. Algebra 55(2), 257–280 (1978)
Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles. EMS Series of Lectures in Mathematics, 2nd edn. European Mathematical Society (EMS), Zürich (2009)
Penttila, T., Williford, J.: New families of \(Q\)-polynomial association schemes. J. Comb. Theory Ser. A 118(2), 502–509 (2011)
Segre, B.: Forme e geometrie hermitiane, con particolare riguardo al caso finito. Annali di Matematica Pura ed Applicata Serie Quarta 70, 1–201 (1965)
Thas, J.A.: \(3\)-regularity in generalized quadrangles: a survey, recent results and the solution of a longstanding conjecture. Rend. Circ. Mat. Palermo (2) Suppl. 53, 199–218 (1998). Combinatorics ’98 (Mondello)
Thas, K.: A stabilizer lemma for translation generalized quadrangles. Eur. J. Comb. 28(1), 1–16 (2007)
van Dam, E.R., Martin, W.J., Muzychuk, M.: Uniformity in association schemes and coherent configurations: cometric \(Q\)-antipodal schemes and linked systems. J. Comb. Theory Ser. A 120(7), 1401–1439 (2013)
van Lint, J.H., Wilson, R.M.: A Course in Combinatorics, 2nd edn. Cambridge University Press, Cambridge (2001)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bamberg, J., Lee, M. A relative m-cover of a Hermitian surface is a relative hemisystem. J Algebr Comb 45, 1217–1228 (2017). https://doi.org/10.1007/s10801-017-0739-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-017-0739-5