Abstract
The space Alt(×3 V 6) of alternating trilinear forms on V 6 = V(6, 2) is naturally isomorphic to the space \({\wedge^{3}(V_{6} ^{\ast})}\) of trivectors based on the dual space \({V_{6}^{\ast}}\). Under the natural action of the group GL(6, 2) the nonzero elements of \({{\rm Alt}(\times^{3}V_{6})\cong\wedge^{3}(V_{6}^{\ast})}\) are shown to fall into five distinct orbits. In consequence, the cubic hypersurfaces in PG(5, 2) are classified into five large families. For \({T \in {\rm Alt}(\times^{3}V_{6})}\) let \({\mathcal{L}_{T}}\) denote the set of T-singular lines, consisting that is of those projective lines \({\langle a,b\rangle}\) in \({{\rm PG}(5,2)=\mathbb{P}V_{6}}\) such that T(a, b, x) = 0 for all \({x\in V_{6}}\). A description is given of the set \({\mathcal{L}_{T}}\) for a representative T of each of the five GL(6, 2)-orbits. In particular, for one of the orbits \({\mathcal{L}_{T}}\) is a Desarguesian line-spread in PG(5, 2).
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Shaw, R. Trivectors and cubics: PG(5, 2) aspects. J. Geom. 99, 167–178 (2010). https://doi.org/10.1007/s00022-011-0060-8
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DOI: https://doi.org/10.1007/s00022-011-0060-8