Tangent sets in finite spaces (Q1923486)

Let \(K\) be a point set in \(PG (n,q)\). A set \(\varepsilon\) is called a tangent set of \(K\) if \(\varepsilon\) is not contained in a hyperplane and the line joining any two points of \(\varepsilon\) is a tangent of \(K\). Let us denote the cardinality of \(K\) by \(k\) and the cardinality of \(\varepsilon\) by \(e\). Some properties of such sets are investigated in the paper. The most important results are the following: (I) Let \(K\) be a nonsingular quadric in \(PG (n,p^r)\), \(p\) an odd prime. (a) If \(p\) does not divide \(n\), then either \(e= n+1\) or \(K\) has not tangent set; (b) If \(p\) divides \(n\), then either \(e= n+2\) or \(K\) has not tangent set. (II) Let \(K\) be a Hermitian surface in \(PG (3,q^2)\), \(q\) odd. (a) There exists a maximal \(\varepsilon\) with \(e= 2q +2\), and the points of \(\varepsilon\) lie on two skew lines; (b) If there exists \(\varepsilon\) such that any four of its points form a tetrahedron \(T\) compatible with \(K\), then \(e \leq 10\) (a rather long definition of such compatibility can be found in the text). (III) In \(P(2,q)\), if the number of lines joining two points of \(K\) is at least \(q+1\), then \(e< (q^2 + q+1)/2\). (IV) In \(P(2, q)\) let \(B (\varepsilon)\) be a blocking set of the lines meeting \(\varepsilon\) in at least two points and let \(L\) be the set of the lines joining at least two points of \(B (\varepsilon)\). If \(e\geq (q^2 + q+1)/2\), then the cardinality of \(L\) is greater than \(q\). (V) Let \(K\) be a set of points in \(PG(n,q)\), \(q>2\), and let \(\varepsilon\) be a tangent set of \(K\). If \(e> \theta(n)/2\), then \(K\) is a hyperplane. Some examples are given.