Veldkamp Spaces of Low-Dimensional Ternary Segre Varieties

Abstract

Making use of the `Veldkamp blow-up' recipe, introduced by Saniga and others (Ann. Inst. H. Poincar\' e D2 (2015) 309) 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 $k \geq 3$ 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 (2013) 986), there are five (explicitly indicated) hyperplane types such that each is the merger 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 $\mathcal{Q}_0^{+}(15,3) \subset {\rm PG}(15,3) \subset \mathcal{V}(S_4(3))$; and, out of them, seven ones 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 discuss embedding of the binary Veldkamp space into the ternary one.