Polarities in finite 2-uniform projective Hjelmslev planes (Q1093159)

Let \(H=(P,B,I)\) be a finite projective 2-uniform Hjelmslev plane subject to the following definition. Any two points (lines) have at least one joining line (one point of intersection), and there is an epimorphism of H onto an ordinary projective plane \(\bar H\) such that any two points (lines) have the same image if and only if they are neighbours (i.e. are incident with more than one common line or point, respectively). 2- uniformity is the following condition. Let L and M be two neighbouring lines passing through p. If q is incident with L and a neighbour of p then q is incident with M. The author studies a polarity \(\pi\) of H, i.e. an involution \(\pi\) : \(P\cup B\to P\cup B\) such that \(\pi (P)=B\), \(\pi (B)=P\) and \(\pi\) preserves incidence and neighbour relation. Estimates for the number of absolute points are given. Example 1. Suppose that H is not an ordinary projective plane and that \(\pi\) has at least one absolute point. Then \(\pi\) has at least \(t^ 2+t\) (at least 2t) absolute points if t is odd (if t is even), where \(t^ 2\) denotes the number of points contained in every neighbour class. Example 2. Suppose that H admits a commutative coordinate ring, t is odd, and \(\Phi =1\), where \(\Phi\) is the involution of the coordinate ring induced by \(\pi\). Furthermore, suppose that the set of absolute points of the induced polarity \({\bar \pi}\) of \(\bar H\) is collinear or an oval. Then the set of absolute points of \(\pi\) is an oval of H consisting of \(t^ 2+t\) points.