Hemisystems on the Hermitian Surface

The natural geometric setting of quadrics commuting with a Hermitian surface of PG(3, q 2 ), q odd, is adopted and a hemisystem on the Hermitian surface H (3, q 2 ) admitting the group P Ω − (4, q ) is constructed, yielding a partial quadrangle PQ(( q −1)/2, q 2 ,( q −1) 2 /2) and a strongly regular graph srg(( q 3 +1)( q +1)/2,( q 2 +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, q 2 ) for q >3, so these are counterexamples to his conjecture. Furthermore, a hemisystem on H (3,25) admitting 3. A 7 .2 is constructed. Finally, special sets (after Shult) and ovoids on H (3, q 2 ) are investigated.