\(k\)-nets embedded in a projective plane over a field (Q343205)

An \textit{(abstract) \(k\)-net} is a point-line incidence structure whose lines are partitioned into \(k\) subsets, called \textit{components}, such that any two lines from distinct components are concurrent with exactly one line from each component. The components have the same size, called the \textit{order} of the \(k\)-net and denoted by \(n\). A \(k\)-net has \(n^2\) points and \(kn\) lines. The affine plane AG\((2,3)\), embedded in the projective plane \(\mathrm{PG}(2,\mathbb{C})\), is a well-known example of a 4-net of order 3. \textit{J. Stipins} [``Old and new examples of \(k\)-nets in \(\mathbb{P}^2\)'', Preprint] has proven that no \(k\)-net, with \(k\geq 5\), exists in \(\mathrm{PG}(2,\mathbb{C})\). This result, and its proof, is in fact true for every projective plane \(\mathrm{PG}(2,\mathbb{K})\), with \(\mathbb{K}\) an algebraically closed field of characteristic zero. The authors reprove this result of J. Stipins [loc. cit.], but they additionally investigate the existence of \(k\)-nets in projective planes \(\mathrm{PG}(2,\mathbb{K})\) over fields \(\mathbb{K}\) of positive characteristic \(p\). This leads to extra examples. Such projective planes \(\mathrm{PG}(2,\mathbb{K})\) over fields \(\mathbb{K}\) of positive characteristic \(p\) contain affine subplanes AG\((2,p)\) of order \(p\), which contain \(k\)-nets, with \(k\leq p+1\). But, no 5-net of order \(n\), with \(p>n\), is known to exist. This suggests that for sufficiently large characteristic \(p\), compared to \(n\), Stipins result [loc. cit.] remains valid in \(\mathrm{PG}(2,\mathbb{K})\). The authors indeed prove this result for characteristic \(p\) large enough. A particular lower bound is, for instance, \(p>3^{n^2/2}\).