Some infinite families of locally partial geometries generalizing the classical Laguerre planes (Q1340111)

An ovoidal Laguerre plane in 3-dimensional projective space \(PG(3,q)\) has as point set the cone over an oval with its vertex \(p\) removed, as generators those lines of \(PG(3,q)\) that pass through \(p\) and as circles the intersections of the cone with those planes of \(PG(3,q)\) that do not pass through \(p\). The author generalizes this construction as follows. Let \(\Omega\) be a firm and residually connected subgeometry of \(PG(n,q)\) whose point set \(O\) generates a \(\beta\)-dimensional subspace \(W\) of \(PG(n,q)\), \(1\leq\beta\leq n - 1\). (\(O\) determines \(\Omega\) as the geometry \(\Gamma^ \beta (\Omega)\) of all subspaces of \(W\) that are generated by their intersection with \(O\).) Furthermore, let \(F\) be a flag of \(PG(n,q)\) consisting of \(i\)-dimensional subspaces \(F_ i\) of \(PG(n,q)\) for \(i = 0,1,\dots, n -\beta - 1\) such that \(F_{n -\beta - 1}\) is disjoint from \(W\). Then \(C_{(F,O)}\) is the cone with basis \(O\) and vertex \(F_{n - \beta - 1}\) with its vertex removed, i.e. \(C_{(F,O)} =\{x\in PG(n,q)|\langle x, F_{n -\beta - 1}\rangle\cap O\neq\emptyset\}\) where \(\langle x,F_{n -\beta - 1}\rangle\) denotes the subspace spanned by \(x\) and \(F_{n -\beta - 1}\). In the case \(n =\beta + 1\), \(C_{(F,O)}\) is just the ordinary cone with vertex the point \(F_ 0\) and base \(O\). The geometry \(\Gamma^{(n,\beta,\Omega)}\) then has point set the points of \(\Omega\); these are the 0-elements of \(\Gamma^{(n,\beta,\Omega)}\). The \(i\)-elements of \(\Gamma^{(n,\beta,\Omega)}\) are the \(i\)-dimensional subspaces of \(PG(n,q)\) that do not intersect \(F_{n - i - 1}\) and whose span with \(F_{n -\beta - 1}\) intersects \(W\) in an \(i\)-element of \(\Omega\), if \(1\leq i\leq\beta - 1\), or which are generated by their intersection with \(C_{(F,O)}\), if \(\beta\leq i\leq n - 1\). Incidence is symmetrized inclusion. It is shown that the \(i\)-elements of \(\Gamma^{ (n,\beta,\Omega)}\), \(\beta\leq i\leq n - 1\), are the nonempty intersections of \(C_{(F,O)}\) and an \(i\)-dimensional subspace disjoint from \(F_{n - i - 1}\). It then follows that \(\Gamma^{(n,\beta,\Omega)}\) is a firm residually connected geometry of rank \(n\) for \(1\leq\beta\leq n\). Furthermore, if \(n\geq\beta + 1\), then the residue of an \((n - 1)\)-element of \(\Gamma^{(n,\beta,\Omega)}\) is isomorphic to \(\Gamma^{(n - 1,\beta,\Omega)}\). Setting \(\Gamma^{(\beta,\beta,\Omega)} =\Omega\), one therefore obtains a series of residually connected geometries of rank \(n\) for each \(n\geq\beta\) each extending the rank \(n - 1\) geometry by a partial geometry. The author also determines the diagrams of the geometries \(\Gamma^{(n,\beta,\Omega)}\) for various subgeometries \(\Omega\) and various \(n\) and \(\beta\). The surrounding projective space \(PG (n,q)\) can be reconstructed from \(\Gamma^{(n, \beta, \Omega)}\) if \(\Omega\) is a nice subgeometry of \(PG (\beta, q)\), \(\beta \geq 2\), that is, for every point \(x\) of \(PG (\beta, q)\) there is a line of \(\Omega\) through \(x\). In this case the automorphism group of \(\Gamma^{(n, \beta, \Omega)}\) equals the subgroup of \(P \Gamma L(n + 1, q)\) consisting of the elements stabilizing \(\Gamma^{(n, \beta, \Omega)}\). Moreover, if the stabilizer of \(\Omega\) in \(PG (\beta + 1, q)\) acts flag-transitively on \(\Omega\), then \(\Gamma^{(n, \beta, \Omega)}\) admits a flag-transitive group of automorphisms.