On Quasi‐Hermitian Varieties

Abstract

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.

Characterizing Hermitian Varieties in Three- And Four-Dimensional Projective Spaces

“…For q = 2, there is just one point set in PG (3,4) up to equivalence, meeting each line in one, three, or five points and each plane in five, nine, or 13 points, that is, the Hermitian cone; see [8,Theorem 19.6.8]. Thus also for q = 2 our theorem follows.…”

“…Consequently, the number of rational points N 4 (S) ∈ {33, 29, 25}. In order to prove that none of the previous possibilities can occur for N 4 (S), we count in a double way the number of planes, the number of pairs (P, π), where P ∈ PG (3,4) and π is a plane through P, and the number of pairs ((P, Q), π), where P, Q ∈ PG(3, 4) and π is a plane through P and Q. Let x, y, z denote the numbers of 5-, 9-, and 13-planes, respectively, we get the following equations:…”

Intersection sets, three-character multisets and associated codes

“…Assume B(a, b) to have Equation (4). It is straightforward to see that B(a, b) coincides with the affine part of the Hermitian variety H of equation (2) in the space Π a ; hence, any hyperplane π P∞ of PG(r, q 2 ) passing through P ∞ meets B(a, b) in |H ∩ π P∞ | = q 2r−3 points. Now we are interested in the possible intersection sizes of B(a, b) with a generic hyperplane π :…”

“…that is the number N of affine points which lie in B(a, b) ∩ π is the same as the number of points of the affine quadric Q of AG(2r−2, q) of Equation (7). Following the approach of [2], in order to compute N, we first count the number of points of the quadric at infinity Q ∞ := Q∩Π ∞ of Q and then we determine N = |Q|−|Q ∞ |.…”

“…Remark 3.3. The quadric Q a (m, d) of Equation (1) shares its tangent hyperplane at P ∞ with the Hermitian variety (2).…”