Quasi‐Hermitian varieties scriptV in PG (r,q2) are combinatorial generalizations of the (nondegenerate) Hermitian variety H(r,q2) so that scriptV and H(r,q2) have the same size and the same intersection numbers with hyperplanes. In this paper, we construct a new family of quasi‐Hermitian varieties. The isomorphism problem for the associated strongly regular graphs is discussed for r=2.
Some new results on plane F q 2 -maximal curves are stated and proved. By [32], the degree d of a plane F q 2 -maximal curve is less than or equal to q + 1 and equality holds if and only if the curve is F q 2 -isomorphic to the Hermitian curve. We show that d ≤ q + 1 can be improved to d ≤ (q + 2)/2 apart from the case d = q + 1 or q ≤ 5. This upper bound turns out to be sharp for q odd. In [4] it was pointed out that some Hurwitz curves are plane F q 2 -maximal curves. Here we prove that (1.3) is the necessary and sufficient condition for a Hurwitz curve to be F q 2 -maximal. We also show that this criterium holds true for the F q 2 -maximality of a wider family of curves.
In a previous paper [1], all point sets of minimum size in PG(2; q), blocking all external lines to a given irreducible conic C C, have been determined for every odd q. Here we obtain a similar classification for those point sets of minimum size, which meet every external and tangent line to C C. #
We investigate the intersection R of two permutable Hermitian surfaces of PG(3, q 2 ), q odd. We show that R is a determinantal variety. From the combinatorial point of view R comprises a complete (q 2 + 1)-span of the two corresponding Hermitian surfaces.
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