On embeddings of minimum dimension of PG(n,q)×PG(n,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{PG}}(n,q

A construction is given of an embedding of \({\mathrm{PG}}(n-1,q)\times {\mathrm{PG}}(n-1,q)\) into \({\mathrm{PG}}(2n-1,q)\), i.e. of minimum dimension, and it is shown that the image is a nonsingular hypersurface of degree \(n\). The construction arises from a scattered subspace with respect to a Desarguesian spread in \({\mathrm{PG}}(2n-1,q)\). By construction there are two systems of maximum subspaces (in this case \((n-1)\)-dimensional) which cover this hypersurface. However, unlike the standard Segre embedding, the minimum embedding constructed here allows another \(n-2\) systems of maximum subspaces which cover this embedding. We describe these systems and study the stabiliser of these embeddings. The results can be considered as a generalization of the properties of the hyperbolic quadric \(Q^+(3,q)\).