On locally polar geometries whose planes are affine (Q913242)

The author deals with geometries \(\Gamma\) belonging to diagrams of the following shape \[ \underset{\underset{X-1} 0} \circ \overset {Af} {\text{---}} \underset{\underset {x} 1} \circ \text{---} \underset{\underset {x} 2} \circ ------ \underset{\underset {x} {n-2}} \circ \text{---} \underset{\underset {x} {n-1}} \circ === \underset{\underset {x} n} \circ \] (n\(\geq 2\); \(1<x<\infty\); \(y<\infty)\) where \(\circ\overset {Af} {\text{---}}\circ\) denotes the class of affine planes, \(x-1\), \(x\) and \(y\) are parameters and 0,1,...,n-1, n are types. Such a geometry is called the \(Af.C_ n\) geometry. Several examples of \(Af.C_ n\) geometries are discussed. Particularly they are the quotient geometries and generalized Fischer spaces, nondegenerated locally classical \(Af.C_ n\) geometries. An important case is a geometry \({\bar \Gamma}\simeq \Gamma /T\) where \(\Gamma\) /T can be realized inside the 2n-dimensional vector space \(V=V(2n,q)\) over GF(q). It is called a holomorph of the symplectic cone (of order q and rank n). The second important case is a geometry \(\Gamma '=\Gamma /\sigma\), \(\sigma\in Aut(\Gamma)\). It is called exceptional geometry for the symplectic group Sp(2m,2). A non-locally polar example, the so called Neumaier geometry \((Af.C_ 3\) geometry), and degenerated example of \(Af.C_ 2\) geometry are given here too. The next part of the article introduces the so called connection index \(\gamma\) and several theorems about properties of \(Af.C_ n\) geometries depending on \(\gamma\). A special part is given to the case \(\gamma =x\), and the following important theorem is proved: We have \(\gamma =x\) if and only if \(\Gamma\) is either the holomorph of a symplectic cone or one of the exceptional geometries \((D_ n^{\pm})/2\) for the symplectic group Sp(2m,2) \((m=n\) or \(n+1)\).