Veldkamp Spaces of Low-Dimensional Ternary Segre Varieties

Making use of the `Veldkamp blow-up' recipe, introduced by Saniga et al. (Ann Inst H Poincare 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 >= 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 2x2x2x2 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 Q(0)(+)(15,3)subset of PG(15,3)subset of 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 W(7,3)subset of 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 V(S-3(2)) are embeddable and one of them, surprisingly, into a non-projective line of V(S-3(3)) only.