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.
Hemisystems on the Hermitian surface
COSSIDENTE, Antonio;
2005-01-01
Abstract
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.File | Dimensione | Formato | |
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