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 Sk(3)\documentclass[12pt]{minimal}
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\begin{document}$$S_k(3)$$\end{document}, where Sk(3)\documentclass[12pt]{minimal}
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\begin{document}$$S_k(3)$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$k=2$$\end{document} such elements are found only among Veldkamp lines, for k≥3\documentclass[12pt]{minimal}
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\begin{document}$$k \ge 3$$\end{document} 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 S3(3)\documentclass[12pt]{minimal}
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\begin{document}$$S_3(3)$$\end{document}, 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 S3(3)\documentclass[12pt]{minimal}
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\begin{document}$$S_3(3)$$\end{document}, whose blowing-ups yield 43 distinct types of projective geometric hyperplanes of S4(3)\documentclass[12pt]{minimal}
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\begin{document}$$S_4(3)$$\end{document}. As the latter number falls short of 48, the number of different large orbits of 2×2×2×2\documentclass[12pt]{minimal}
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\begin{document}$$2 \times 2 \times 2 \times 2$$\end{document} 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 S4(3)\documentclass[12pt]{minimal}
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\begin{document}$$S_4(3)$$\end{document}, featuring 7,176,640 members in total, that are in a one-to-one correspondence with the points lying on the unique hyperbolic quadric Q0+(15,3)⊂PG(15,3)⊂V(S4(3))\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {Q}_0^{+}(15,3) \subset \mathrm{PG}(15,3) \subset \mathcal {V}(S_4(3))$$\end{document}; and, out of them, seven types that correspond bijectively to the set of 91,840 generators of the symplectic polar space W(7,3)⊂V(S3(3))\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {W}(7,3) \subset \mathcal {V}(S_3(3))$$\end{document}. For k=3\documentclass[12pt]{minimal}
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\begin{document}$$k=3$$\end{document} 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(S3(2))\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {V}(S_3(2))$$\end{document} are embeddable and one of them, surprisingly, into a non-projective line of V(S3(3))\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {V}(S_3(3))$$\end{document} only.
