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Conic sections in space defined by intersection conditions. (English) Zbl 1090.51007
The author continues the study of the set $$S_m$$, $$0\leq m\leq 3$$, of planes in the projective three-space that intersect $$m$$ conic sections and $$6-2m$$ straight lines in six points of a conic. The paper under review deals with the cases $$m=2, 3$$. (See Beitr. Algebra Geom. 46, No. 2, 435–446 (2005; Zbl 1090.51008) for $$m=0$$ and J. Geom. Graph. 8, 59–68 (2004; Zbl 1081.51501) for $$m=1$$.) By using a unified method, it is shown that for all $$S_m$$, $$0\leq m\leq 3$$, the solution manifold is in general algebraic, of class $$8-m$$. The algebraic equation for $$S_m$$ in the homogeneous plane coordinates is derived from a non-algebraic equation for a supermanifold. Special lines and planes in $$S_m$$ are also determined.
##### MSC:
 51N15 Projective analytic geometry 51N35 Questions of classical algebraic geometry 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
##### Keywords:
incidence; conic section; pencil of planes; star product
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