Quadrics as hyperplanes in finite affine geometries (Q1195344)

The author investigates finite affine spaces \(AG(n,q)\), \(q\) odd. This is a continuation of his previous article concerning projective planes \(PG(n,q)\), \(n\) even, \(q\) odd [ibid. 48, 303-313 (1982; Zbl 0503.51009)]. An analogous result is obtained. The main theorem is the following: In a finite Desarguesian affine geometry \({\mathcal A}=({\mathcal P}, {\mathcal L}, {\mathcal I})\) of odd order \(q\), there exists a family of affine quadrics which correspond in a natural way to the hyperplanes of \({\mathcal A}\), and whose intersections provides a family \({\mathcal L}'\) of \(q\)-sets, and an incidence relation \({\mathcal I}'\subset{\mathcal P}\times{\mathcal L}'\), such that \({\mathcal A}'=({\mathcal P},{\mathcal L}',{\mathcal I}')\) is an affine geometry isomorphic to \({\mathcal A}\).