HEMISYSTEMS ON THE HERMITIAN SURFACE

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 PΩ−(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.