k-nets embedded in a projective plane over a field

Abstract

We investigate k-nets with k≥4 embedded in the projective plane PG(2,$\mathbb{K}$) defined over a field $\mathbb{K}$; they are line configurations in PG(2,$\mathbb{K}$) consisting of k pairwise disjoint line-sets, called components, such that any two lines from distinct families are concurrent with exactly one line from each component. The size of each component of a k-net is the same, the order of the k-net. If $\mathbb{K}$ has zero characteristic, no embedded k-net for k≥5 exists; see [11,14]. Here we prove that this holds true in positive characteristic p as long as p is sufficiently large compared with the order of the k-net. Our approach, different from that used in [11,14], also provides a new proof in characteristic zero.