Geometric structures arising from generalized -planes

Abstract

We study translation planes constructed by Andr\'e net replacement on -planes and derivation on -planes. Then, we get to the conclusion that the family of non-Andr\'e -planes is new, and thus so are their replaced and derived planes.
We also study a new way to construct translation planes by putting together two `halves' of planes that belong to two different -planes. We show examples of planes of small order constructed this way.
Finally, we prove that using regular hyperbolic covers, -planes induce partitions of Segre varieties by Veronesians (sometimes called flat flocks)