To characterize Hermitian varieties in projective space PG(d, q) of d dimensions over the Galois field GF(q), it is necessary to find those subsets K for which there exists a fixed integer n satisfying (i) 3 ⩽ n ⩽ q − 1, (ii) every line meets K in 1, n or q + 1 points. K is called singular or non-singular as there does or does not exist a point P for which every line through P meets K in 1 or q + 1 points. For q odd, a non-singular K is a non-singular Hermitian variety (M. Tallini Scafati “Caratterizzazione grafica delle forme hermitiane di un Sr, q” Rend. Mat. Appl.26 (1967), 273–303). For q even, q > 4 and d = 3, a non-singular K is a Hermitian surface or “looks like” the projection of a non-singular quadric in PG(4, q) (J.W.P. Hirschfeld and J.A. Thas “Sets of type (1, n, q + 1) in PG(d, q)” to appear). The case q = 4 is quite exceptional, since the complements of these sets K form a projective geometry code, a (21, 11) code for d = 2 and an (85, 24) code for d = 3. The full list of these sets is given.
