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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Bamberg J., Betten A., Cara P., De Beule J., Lavrauw M., Neunhoeffer M.: FinInG–a GAP package for finite incidence geometry, 1.01. http://cage.ugent.be/geometry/fining (2012). Accessed 29 July 2013.
Bichara A., Misfeld J., Zanella C.: Primes and order structure in the product spaces. J. Geom. 58, 53–60 (1997).
Bichara A., Havlicek H., Zanella C.: On linear morphisms of product spaces. Discret. Math. 267, 35–43 (2003).
Blokhuis A., Lavrauw M.: Scattered spaces with respect to a spread in \({{\rm PG}}(n, q)\). Geom. Dedicata 81, 231–243 (2000).
Huppert B.: Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Band 134. Springer, Berlin (1967).
Kantor W.M.: Linear groups containing a Singer cycle. J. Algebra 62, 232–234 (1980).
Lavrauw M.: Scattered spaces with respect to spreads, and eggs in finite projective spaces. Dissertation, Eindhoven University of Technology, Eindhoven, pp. viii + 115 (2001).
Lavrauw M.: On the isotopism classes of finite semifields. Finite Fields Appl. 14, 897–910 (2008).
Lavrauw M.: Finite semifields with a large nucleus and higher secant varieties to Segre varieties. Adv. Geom. 11, 399–410 (2011).
Lavrauw M., Sheekey J.: Orbits of the stabiliser group of the Segre variety product of three projective lines. Preprint (2012).
Lavrauw M., Van de Voorde G.: On linear sets on a projective line. Des. Codes Cryptogr. 56, 89–104 (2010).
Lavrauw M., Van de Voorde G.: Scattered linear sets and pseudoreguli. Electron. J. Comb. 20(1), Research Paper P15 (2013).
Lavrauw M., Zanella C.: Segre embeddings and finite semifields. Finite Fields Appl. (2013). doi:10.1016/j.ffa.2013.07.005.
Lavrauw M., Marino G., Polverino O., Trombetti R.: \({\mathbb{F}}_q\)-pseudoreguli of \({{\rm PG}}(3, q^3)\) and scattered semifields of order \(q^6\). Finite Fields Appl. 17, 225–239 (2011).
Lunardon G., Polverino O.: Blocking sets and derivable partial spreads. J. Algebraic Comb. 14, 49–56 (2001).
Lunardon, G. Marino, G. Polverino O., Trombetti R.: Maximum scattered linear sets of pseudoregulus type and the Segre Variety \({\cal {S}}_{n, n}\). Preprint. arXiv:1211.3604 [math.CO].
The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.6.2. http://www.gap-system.org (2013). Accessed 29 July 2013.
Zanella C.: Universal properties of the Corrado Segre embedding. Bull. Belgian Math. Soc. Simon Stevin 3, 65–79 (1996).
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