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 \(Q^+(3,q)\).
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Acknowledgments
The authors thank Rod Gow for his helpful remarks in the preparation of this paper. This research was supported by a Progetto di Ateneo from Università di Padova (CPDA113797/11). The first author acknowledges the support from FWO-Flanders.
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Communicated by C. Mitchell.
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Lavrauw, M., Sheekey, J. & Zanella, C. On embeddings of minimum dimension of \({\mathrm{PG}}(n,q)\times {\mathrm{PG}}(n,q)\) . Des. Codes Cryptogr. 74, 427–440 (2015). https://doi.org/10.1007/s10623-013-9866-8
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DOI: https://doi.org/10.1007/s10623-013-9866-8