Smooth stable planes and the moduli spaces of locally compact translation planes

Smooth stable planes have been introduced in [3]. At every point p of a smo oth stable plane S = (P, L, F) the tangent spaces of the lines through p fo rm a compact spread (see the definition in Section 2) on the tangent space TpP thus defining a locally compact topological affine translation plane A( p). We introduce the moduli space TpP of isomorphism classes of compact spr eads, l is an element of {1, 2, 4, 8}. We show that for l > 1 the topology of J(l)(R-2l) is not T-1 by constructing a sequence of non-classical spread s in F-2 that converges to the classical spread in F-2 where F is an elemen t of {C, H, O}. Moreover. we prove that the isomorphism type of A(p) varies continuously with the point p. Finally, we give examples of smooth affine planes which have both classical and non-classical tangent translation plan es.