Embedding the planar structure of 4-dimensional linear spaces into projective spaces (Q1288019)

A \(d\)-dimensional linear space \(L\) is called the direct product of two its subspaces \(X\) and \(Y\), if the dimensions of \(X\) and \(Y\) sum up to \(d-1\) and if every point lies in one (and thus exactly one) of these subspaces. If \(L\) is the direct product of two non-trivial subspaces, then we call \(L\) decomposible. It is known that every indecomposible linear space has at least as many hyperplanes as points with equality if and only if it is a projective space. Theorem 1.1. Let \(L\) be an indecomposible 4-dimensional linear space with \(v\) points and \(h\) hyperplanes. Suppose that \(m^4\leq v\leq h\leq m^4+ \delta m^3\) where \(m>10^{11}\) and \(\delta=10^{-5}\sqrt{m}\). Then \(L_3\) (three-dimensional structure of a space \(L\)) embeds in \(PG(4,q)_3\).