# Veldkamp Spaces of Low-Dimensional Ternary Segre Varieties

- 24 Downloads

## Abstract

Making use of the ‘Veldkamp blow-up’ recipe, introduced by Saniga et al. (Ann Inst H Poincaré D 2:309–333, 2015) for binary Segre varieties, we study geometric hyperplanes and Veldkamp lines of Segre varieties \(S_k(3)\), where \(S_k(3)\) stands for the *k*-fold direct product of projective lines of size four and *k* runs from 2 to 4. Unlike the binary case, the Veldkamp spaces here feature also non-projective elements. Although for \(k=2\) such elements are found only among Veldkamp lines, for \(k \ge 3\) they are also present among Veldkamp points of the associated Segre variety. Even if we consider only projective geometric hyperplanes, we find four different types of non-projective Veldkamp lines of \(S_3(3)\), having 2268 members in total, and five more types if non-projective ovoids are also taken into account. Sole geometric and combinatorial arguments lead to as many as 62 types of projective Veldkamp lines of \(S_3(3)\), whose blowing-ups yield 43 distinct types of projective geometric hyperplanes of \(S_4(3)\). As the latter number falls short of 48, the number of different large orbits of \(2 \times 2 \times 2 \times 2\) arrays over the three-element field found by Bremner and Stavrou (Lin Multilin Algebra 61:986–997, 2013), there are five (explicitly indicated) hyperplane types such that each is the fusion of two different large orbits. Furthermore, we single out those 22 types of geometric hyperplanes of \(S_4(3)\), featuring 7,176,640 members in total, that are in a one-to-one correspondence with the points lying on the unique hyperbolic quadric \(\mathcal {Q}_0^{+}(15,3) \subset \mathrm{PG}(15,3) \subset \mathcal {V}(S_4(3))\); and, out of them, seven types that correspond bijectively to the set of 91,840 generators of the symplectic polar space \(\mathcal {W}(7,3) \subset \mathcal {V}(S_3(3))\). For \(k=3\) we also briefly discuss embedding of the binary Veldkamp space into the ternary one. Interestingly, only 15 (out of 41) types of lines of \(\mathcal {V}(S_3(2))\) are embeddable and one of them, surprisingly, into a *non*-projective line of \(\mathcal {V}(S_3(3))\) only.

## Keywords

Ternary Segre varieties Veldkamp spaces finite polar spaces## Mathematics Subject Classification

51A45 51E20 68R05## Notes

### Acknowledgements

This work was supported by the Slovak Research and Development Agency under the contract # SK-FR-2017-0002 and the French Ministry of Europe and Foreign Affairs (MEAE) under the Project PHC Štefánik 2018/40494ZJ. The financial support of both the Slovak VEGA Grant Agency, Project # 2/0003/16, and the French “Investissements d’Avenir” programme, project ISITE-BFC (Contract ANR-15-IDEX-03), are gratefully acknowledged as well. We are also indebted to Dr. Petr Pracna for the help with the figures and thank an anonymous referee for helpful comments.

## References

- 1.Buekenhout, F., Cohen, A.M.: Diagram Geometry: Related to Classical Groups and Buildings. Springer, Berlin (2013). Sec. 8.2CrossRefGoogle Scholar
- 2.Cohen, A.M.: Point-line spaces related to buildings. In: Buekenhout, F. (ed.) Handbook of Incidence Geometry, pp. 647–737. Elsevier, Amsterdam (1995)CrossRefGoogle Scholar
- 3.Saniga, M., Lévay, P., Planat, M., Pracna, P.: Geometric hyperplanes of the near hexagon L\(_3 \times \)GQ(2, 2). Lett. Math. Phys.
**91**, 93–104 (2010)MathSciNetCrossRefGoogle Scholar - 4.Saniga, M., Havlicek, H., Holweck, F., Planat, M., Pracna, P.: Veldkamp-space aspects of a sequence of nested binary Segre varieties, Annales de l’Institut Henri Poincaré D 2, 309–333 (2015). see also arXiv:1403.6714
- 5.Hirschfeld, J.W.P., Thas, J.A.: General Galois Geometries. Oxford University Press, Oxford (1991)zbMATHGoogle Scholar
- 6.Shult, E.E.: Points and Lines: Characterizing the Classical Geometries. Springer, Berlin (2011)CrossRefGoogle Scholar
- 7.De Beule, J., Klein, A., Metsch, K.: Substructures of finite classical polar spaces. In: De Beule, J., Storme, L. (eds.) Current Research Topics in Galois Geometry, pp. 33–59. Nova Science Publishers, New York (2011)Google Scholar
- 8.Bremner, M.R., Stavrou, S.: Canonical forms of \(2 \times 2 \times 2\) and \(2 \times 2 \times 2 \times 2\) arrays over \(F_2\) and \(F_3\). Linear Multilinear Algebra
**61**, 986–997 (2013). see also arXiv:1112.0298 MathSciNetCrossRefGoogle Scholar - 9.Dyck, W.: Über Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer Riemann’sher Flächen. Math. Ann.
**17**, 473–509 (1881)MathSciNetCrossRefGoogle Scholar - 10.Eppstein, D.: The many faces of the Nauru graph, published online at http://11011110.livejournal.com/124705.html (2016). Accessed Oct 23 2016
- 11.Green, R.M., Saniga, M.: The Veldkamp space of the smallest slim dense near hexagon. Int. J. Geom. Methods Mod. Phys.
**10**, 1250082 (2013)MathSciNetCrossRefGoogle Scholar - 12.Holweck, F., Saniga, M., Lévay, P.: A notable relation between \(N\)-qubit and \(2^{N-1}\)-qubit Pauli groups via binary \(LGr (N, 2N)\). Symmetry Integr. Geom. Methods Appl.
**10**, 041 (2014)zbMATHGoogle Scholar