The affine plane \(AG(2,q)\), \(q\) odd, has a unique one point extension (Q1340651)

Let \({\mathcal I}\) be an inversive plane of odd order \(q\), i.e. a \(3 - (q^ 2 + 1, q + 1,1)\) design. If \(p\) is a point of \(\mathcal I\), then the internal plane \({\mathcal I}_ p\) of \(\mathcal I\) at \(p\) is defined as follows: The points of \({\mathcal I}_ p\) are the points of \(\mathcal I\) different from \(p\), and the lines of \({\mathcal I}_ p\) are the blocks of \(\mathcal I\) containing \(p\) (minus \(p\)). The internal plane \({\mathcal I}_ p\) is an affine plane of order \(q\). Let \(q\) be a prime power, and denote by \(O\) an elliptic quadric of \(\text{PG}(3,q)\). The points of \(O\) together with the intersections \(\pi \cap O\), with \(\pi\) a non-tangent plane of \(O\), form an inversive plane of order \(q\). An inversive plane which is isomorphic to such an example is called Miquelian. In the paper under review, the author proves the following theorem: If the internal plane of \(\mathcal I\) at \(p\) is Desarguesian for some point \(p\) of \(\mathcal I\), then the inversive plane \(\mathcal I\) is Miquelian. Under some additional restrictions on \(q\), the author has already proved this result in Contemp. Math. 111, 187-218 (1990; Zbl 0728.51010) and `Finite geometries, buildings, and related topics', Pap. Conf., Pingree Park/CO (USA) 1988, 145-159 (1990; Zbl 0754.51007). In the present paper, he gives a new, uniform proof for all odd values of \(q\). Choose a point \(p\) of \(\mathcal I\) such that the internal plane \({\mathcal I}_ p\) is Desarguesian. The blocks of \(\mathcal I\) through \(p\) are the lines of \({\mathcal I}_ p\), and the remaining blocks of \(\mathcal I\) are \((q + 1)\)-arcs of \({\mathcal I}_ p\). The author shows that this set of \((q + 1)\)-arcs of \({\mathcal I}_ p \simeq \text{AG}(2,q)\) can also be obtained from the Miquelian inversive plane, by the same process. The main auxiliary result which is used, is the classification of the flocks of the hyperbolic quadric in \(\text{PG}(3,q)\) [\textit{L. Bader} and \textit{G. Lunardon}, Geom. Dedicata 29, No. 2, 177-183 (1989; Zbl 0673.51010)].