We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Trivectors and cubics: PG(5, 2) aspects.
- Authors
Shaw, Ron
- Abstract
The space Alt(× V) of alternating trilinear forms on V = 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).
- Subjects
MATHEMATICAL singularities; VECTOR analysis; CUBES; TRILINEAR forms; TOPOLOGICAL spaces; ISOMORPHISM (Mathematics); DUALITY theory (Mathematics); PROJECTIVE geometry
- Publication
Journal of Geometry, 2010, Vol 99, Issue 1/2, p167
- ISSN
0047-2468
- Publication type
Article
- DOI
10.1007/s00022-011-0060-8