Characterization of Finite Hermitian Semi-Quadrics

Abstract

Hermitian semi-quadrics of PG (d,q) are characterized as sets of class (0, l,n,q + l) having sufficiently large cardinality. Moreover a complete classification of sets of class (0, 1, n, q+ 1) in P G ( 3 , q) is achieved. Definitions and preliminary resultsLet P = PG {d, q) be the finite projective space of dimension d ^ 2 and order q > 2. Let <X> be the subspace spanned by some subset X of points of P.If Q is a subset of points of P and if n is an integer such that 2 ^ n ^ q, we say that Q is a set of class (0, l , n , g + l) if every line of P meets Q in 0 , 1 , w or g + 1 points. If, furthermore, each line meets Q in at least one point, then we say more precisely that Q is of class (\,n,q+l). Sets of class (0,1, n) of PG (2, a), having more than one point, are special {k, «}-arcs. Recall that a {&,«}-arc in PG (2, q) is a set K of k points of PG (2, q) such that some line of the plane meets K in n points, but such that no line meets K in more than n points.Two distinct points a, b are adjacent if the line <a, fr> is contained in Q; we write a ~ b and we assume that each point is adjacent to itself. Q is degenerate if there is a point that is adjacent to all points of Q. A line of Q is & line which is contained in Q. A secanf line is a line that is not a line of Q and meets Q in more than one point. A line is exterior to Q if it does not intersect Q.It is obvious that if Q is a set of class (0,1, n, q +1) in P and if S is some subspace of P, then S n Q is a set of class (0,1, n, q +1) of S. Let us recall that degenerate sets can be described from non-degenerate ones: see [6]. Recall also that [9, Theorem III] implies the following well-known classification of sets of class (0, l,n,q + l) in a plane. THEOREM 0. IfQ is a set of class (0, l,n,q + l) spanning PG (2, q) and admitting some secant line, then Q is one of the following:(i) the union of n lines having a common point;(ii) the complement of a maximal {(q -n){q + l)+l,q -n + l}-arc;(iii) the union of a maximal {(n -2)(<gr-h 1)+ 1, n -\}-arc and of a line exterior to this arc;(iv) a {k, ri)-arc of class (0,1, ri).Sets of type (iii) can exist only if M -1 divides q. So, putting q = p h , we have n = p ' + l , with 0 < / < h. Similarily, if sets of type (ii) exist, then