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) .
On embeddings of minimum dimension of PG(n,q)×PG(n,q)
LAVRAUW, MICHEL;SHEEKEY, JOHN FRANCIS;ZANELLA, CORRADO
2015
Abstract
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) .
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