Conic sections in space defined by intersection conditions. (English) Zbl 1090.51007

Schröcker, Hans-Peter

Beitr. Algebra Geom. 46, No. 2, 301-309 (2005).

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.

Reviewer: Mihai Cipu (Bucureşti)

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

Citations:

Zbl 1081.51501; Zbl 1090.51008