Hermitian Varieties in a Finite Projective Space PG(N, q²)

Abstract

The geometry of quadric varieties (hypersurfaces) in finite projective spaces of N dimensions has been studied by Primrose (12) and Ray-Chaudhuri (13). In this paper we study the geometry of another class of varieties, which we call Hermitian varieties and which have many properties analogous to quadrics. Hermitian varieties are defined only for finite projective spaces for which the ground (Galois field) GF(q2) has order q2, where q is the power of a prime. If h is any element of GF(q2), then = hq is defined to be conjugate to h.

Szemerédi–Trotter-type theorems in dimension 3

“…Linear spaces on such Hermitian hypersurfaces have been studied in detail [Seg65,BC66]. These examples have also long been recognized as extremal for the Gauss map.…”

“…All of these are easy to do by hand as in see [Seg65,BC66] or can be obtained from the general description of finite unitary groups; see for instance [Car72].…”

The weight distribution of the functional codes defined by forms of degree 2 on Hermitian surfaces

“…(ii) We also know from [2], theorem 7.3 p. 1172, that there are exactly #X(F q ) = t 5 + t 3 + t 2 + 1 tangent planes to the Hermitian surface X. We want now to determine the number of quadrics Q = P 1 ∪ P 2 where P 1 and P 2 are two distinct tangent planes meeting at a line contained in X.…”

MutuallyM-intersecting Hermitian Varieties

“…This means that there exists a nonsingular matrix P such that P2HP O "I for any nonsingular Hermitian matrix H. When H is a Hermitian matrix of rank r, then <(H) is called an HV of rank r. An HV of rank n#1 in PG(n, q) is said to be nondegenerate. The properties of an HV in PG(2, q) have been studied in [1,8]. An HV in PG(2, q) contains q#1, q#q#1, or q#1 points, accordingly as the rank is 1, 2, or 3.…”

“…(1) For aO>"1, the "rst row of P2H H P O is expressed by p2" (1,0,a#cuHO). Then the (1, 1)-entry of (P2H…”