Smooth stable planes and the moduli spaces of locally compact translation planes (Q1568797)

A stable plane consists of a locally compact space \(P\) of points and a collection \(L\) of closed subsets (called lines) such that two points uniquely determine a line, and that the resulting operations of joining and intersecting are continuous (with respect to the Hausdorff topology on~\(L\)) with open domain of definition. A stable plane is called smooth if \(P\) and \(L\) carry smooth structures such that the operations are smooth. Prominent examples are the projective planes over the real numbers, the complex numbers, Hamilton's quaternions or Cayley's octonions. Many other examples appear as deformations of these classical planes, or of open subgeometries. For every point \(p\) of a smooth stable plane, the tangent space \(T_pP\) carries the structure of a locally compact translation plane \(A_p\), obtained by André's procedure from the partition of \(T_pP\) into tangent spaces of lines through~\(p\). These partitions are compact subsets of the Grassmann manifold \(G_l(T_pP)\), where \(l\) denotes the dimension of a line. This yields a topology on the set of locally compact translation planes defined on~\(T_pP\). Calling two partitions equivalent if they yield isomorphic planes, one obtains a quotient of the space of translation planes which is called the moduli space \(J_l(T_p)\). The authors obtain the following. For \(l>1\), the moduli space \(J_l(T_p)\) does not even satisfy the separation axiom~\(T_1\). The tangent translation plane \(A_p\) depends continuously on~\(p\). There exist smooth affine planes containing points with nonisomorphic tangent translation planes.