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Designs, Codes and Cryptography

Erschienen in:

01.02.2015

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

verfasst von: Michel Lavrauw, John Sheekey, Corrado Zanella

Erschienen in: Designs, Codes and Cryptography | Ausgabe 2/2015

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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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Titel: On embeddings of minimum dimension of
verfasst von: Michel Lavrauw
John Sheekey
Corrado Zanella
Publikationsdatum: 01.02.2015
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 2/2015
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-013-9866-8

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

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