On embeddings of minimum dimension of $${\mathrm{PG}}(n,q)\times {\mathrm{PG}}(n,q)$$ PG ( n , q ) × PG ( n , q )

Abstract

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 $$Q^+(3,q)$$ .