Geometric structures arising from generalized $j$-planes

Abstract

We study translation planes constructed by Andr\'e net replacement on $jj\cdots j$-planes and derivation on $jj\cdots j$-planes. Then, we get to the conclusion that the family of non-Andr\'e $jj\cdots j$-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 $jj\cdots j$-planes. We show examples of planes of small order constructed this way.
Finally, we prove that using regular hyperbolic covers, $jj\cdots j$-planes induce partitions of Segre varieties by Veronesians (sometimes called flat flocks)