Intersections of smooth polar unitals with secants are homeomorphic to spheres

Abstract.

In [5] and [6] we proved that (non-empty) sets of absolute points of smooth polarities, i.e. smooth polar unitals, in smooth projective planes of dimension 2l are smooth submanifolds of the point spaces homeomorphic to spheres of dimension $2l - 1 {\rm or} \frac{3}{2}l - 1$. In this paper we show that the intersections of smooth polar unitals with secants are homeomorphic to spheres of dimension $l - 1 {\rm or} \frac{1}{2} l - 1$, respectively. Furthermore we prove that the condition of connectedness in [6, Theorem 1.2] may be omitted. This means that a closed (not necessarily connected) submanifold U of the point space of a smooth projective plane is homeomorphic to a sphere provided that there exists precisely one tangent at each point of U, and each secant intersects U transversally. If U has codimension 1 in the point space then the second condition follows from the first one, and also the intersections of U with secants are homeomorphic to spheres. This result may be generalized to compact hypersurfaces in the point spaces of smooth affine planes.