On embeddings of minimum dimension of PG(n,q)×PG(n,q)

A construction is given of an embedding of PG(n−1,q)×PG(n−1,q) into 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 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).