Arcs and Blocking Sets II

In a finite projective plane 1t a blocking set is a set S of points such that each line contains at least one point in S and at least one point not in S. The main results in this note are Theorems 1.1 and 1.7 and Corollaries 1.8 and 1.9. Theorem 1.1 describes new bounds on certain kinds of reduced blocking sets in PG(2, q). Theorem 1.7 and Corollary 1.8 give new bounds on the cardinality of a reduced (and so, of an arbitrary) blocking set Sin PG(2, q). These bounds yield a significant improvement on previously known results. The proof of 1.7 uses a combinatorial argument together with special cases of some deep results in Jamison [9] and R6dei [10]. A fortuitous factorization makes the result more tractable. (The result of 1.1 is used in 1.4 to obtain bounds on the size of complete arcs in PG(2, q): in some cases these results give a slight improvement on the results in Hirschfeld [8]). Corollary 1.9 shows how the structure ofPG(2, q) is being utilized: it yields a far stronger result than a related result for general planes in Bruen and Thas [5] (the case n > 4 in Theorem 3 there). For blocking sets in arbitrary finite projective planes not much is known apart from 2.2. Here we offer a new proof based on an idea in Hill and Mason [7]. Moreover the proof can be generalized to arbitrary 2-designs as in Theorem 2.3. Fundamental to the improved bound on lSI in PG(2, q) is a result of Jamison [9] on intersection sets in the classical affine plane AG(2, q). His result is not valid for general finite affine planes: the problem of finding good bounds on the size of blocking sets in finite affine planes is open. Our result (Theorem 3.1) describes the best known such bounds. We use the fact that an affine plane (which is of course a 2-design) also has its lines arranged into parallel classes, so our result is a slight improvement on Theorem 2.3 in the case of finite affine planes.