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Journal of the London Mathematical Society 2005 72(3):731-741; doi:10.1112/S0024610705006964
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© The London Mathematical Society

Hemisystems on the Hermitian Surface

Antonio Cossidente and Tim Penttila

Dipartimento di Matematica, Università della Basilicata Contrada Macchia Romana, I-85100 Potenza, Italy cossidente{at}unibas.it
Department of Mathematics and Statistics, University of Western Australia Australia 6009 penttila{at}maths.uwa.edu.au

Received 27 August 2004. Revision received 8 March 2005.

The natural geometric setting of quadrics commuting with a Hermitian surface of PG(3,q2), q odd, is adopted and a hemisystem on the Hermitian surface H(3,q2) admitting the group P{Omega}–(4,q) is constructed, yielding a partial quadrangle PQ((q–1)/2, q2,(q–1)2/2) and a strongly regular graph srg((q3+1)(q+1)/2,(q2+1)(q–1)/2,(q–3)/2,(q–1)2/2). For q>3, no partial quadrangle or strongly regular graph with these parameters was previously known, whereas when q=3, this is the Gewirtz graph. Thas conjectured that there are no hemisystems on H(3,q2) for q>3, so these are counterexamples to his conjecture. Furthermore, a hemisystem on H(3,25) admitting 3.A7.2 is constructed. Finally, special sets (after Shult) and ovoids on H(3,q2) are investigated.


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