Blocking sets in Desarguesian affine and projective planes
A blocking set in a projective or affine plane is a set of points which intersects every line. A blocking set which contains a line is called trivial, while one having no proper subset which is a blocking set is called minimal (or irreducible). While a considerable amount of work has been done in this area, there are still several interesting open questions concerning the spectrum of sizes of minimal non-trivial blocking sets. In this paper the author is mainly concerned with ``small'' blocking sets, i.e., those minimal non- trivial blocking sets whose size lies in the interval (\(q+1, 3(q+1)/2\)), where \(q\) is the order of the finite field over which the projective or affine plane is defined. For odd \(q\) the projective triangle provides an example of size \(3(q+1)/2\), while for even \(q\) the similar projective triad gives an example of size \((3q+2)/2\). Blokhuis has shown that when \(q\) is an odd prime, there is no smaller example than the projective triangle. The known examples in this range are all of Rédei type, that is, if \(B\) is such a blocking set, then there exists a line \(\ell\) which intersects \(B\) in exactly \(| B |- q\) points. The main result of the paper is that a non-trivial minimal blocking set of size less than \(3(q+1)/2\) in a Desarguesian projective plane of order \(q = p^n\) intersects every line in 1 modulo \(p\) points. It is also shown that the size of such a blocking set must lie in a few relatively short (with respect to \(q\)) subintervals. In particular, when \(q = p^2\), these results imply that a non-trivial minimal blocking set either contains a Baer-subplane or has size at least \(3(q+1)/2\), and this result is sharp. The technique used to obtain these results is based on the Rédei polynomial which encodes the sizes of the intersections of lines with a set of points as multiplicities of its roots. Several investigators have studied this polynomial algebraically to obtain information about blocking sets, especially those of Rédei type. Szőnyi, however, in this paper, examines the polynomial from the algebraic geometry point of view. This permits him to investigate more general blocking sets and also obtain alternate proofs of several well known theorems, including the Jamison, Brouwer-Schrijver theorem on blocking sets in Desarguesian affine planes and the above-mentioned result of Blokhuis.
scientific article; zbMATH DE number 1199978
scientific article