\(8_3\) in \(\mathrm{PG}(2,q)\) (Q1077704)

Let \(\mathfrak C\) be a tactical configuration \(8_ 3\) embedded in a projective plane \({\mathfrak P}\). Then \(\mathfrak C\) is in fact isomorphic to \(\mathrm{BAG}(2,3)\), the biaffine plane of order 3 obtained by puncturing the affine plane of order 3. The main result is as follows: \(\mathfrak C\) can be embedded into a Pappian plane \(\mathfrak P=\mathrm{PG}(2,F)\) iff one of the following three cases holds: (i) \(\mathrm{char}\, F=3\); (ii) \(\mathrm{char}\, F=2\) and \(F\) contains a primitive 3rd root of unity; (iii) \(F\) contains a primitive sixth root of unity. In the finite case, this means \(F=\mathrm{GF}(q)\) with \(q\) a power of 3 or 4 or with \(q\equiv 1\bmod 6\). We also show, that each \(8_3\) embedded in \(\mathrm{PG}(2,F)\) extends to an affine plane of order 3 embedded in \(\mathrm{PG}(2,F)\). Thus our result strengthens the theorem of Ostrom and Sherk on embedding \(\mathrm{AG}(2,3)\) into \(\mathrm{PG}(2,q)\) [see \textit{T. G. Ostrom} and \textit{F. A. Sherk}, Can. Math. Bull. 7, 549--559 (1964; Zbl 0125.38401)]. We also compute the number of configurations \(8_3\) (and of affine planes \(\mathrm{AG}(2,3))\) embedded in \(\mathrm{PG}(2,q)\). Particular attention is given to the cases \(q=3\) and \(q=4\).