# Veldkamp Spaces of Low-Dimensional Ternary Segre Varieties

## 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.

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