Veldkamp Spaces of Low-Dimensional Ternary Segre Varieties View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2019-03

AUTHORS

Jérôme Boulmier, Frédéric Holweck, Maxime Pinard, Metod Saniga

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 Sk(3), where Sk(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 S3(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 S3(3), whose blowing-ups yield 43 distinct types of projective geometric hyperplanes of S4(3). As the latter number falls short of 48, the number of different large orbits of 2×2×2×2 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 S4(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 Q0+(15,3)⊂PG(15,3)⊂V(S4(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)⊂V(S3(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(S3(2)) are embeddable and one of them, surprisingly, into a non-projective line of V(S3(3)) only. More... »

PAGES

54

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URI

http://scigraph.springernature.com/pub.10.1007/s00025-019-0974-2

DOI

http://dx.doi.org/10.1007/s00025-019-0974-2

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1112112870


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