Blocking sets, k-arcs and nets of order ten

Our first main objective here is to unify two important theories in finite geometries, namely, the theories of k-arcs and blocking sets. This has a number of consequences, which we develop elsewhere. However, one consequence that we do discuss here is an improvement of Bruck's bound [1] concerning the possibility of embedment of finite nets of order n, in the controversial case when n = 10. The argument also makes use of a recent computer result of Denniston [5]. The second (related) main result involves a new combinatorial bound concerning blocking sets (Theorem 5). We are able to show that the bound is sharp by constructing a new class of geometrical examples of blocking sets in Theorem 6. See also the note added in proof.