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.