Geometry of Projective Planes over Two-Dimensional Algebras

Abstract

The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds X^3 of rank 2. The authors study focal properties of these submanifolds and prove that they represent examples of different types of tangentially degenerate submanifolds. Namely, the submanifold X^3, corresponding in RP^5 to a smooth line γof the projective plane C, does not have real singular points, the submanifold X^3, corresponding in RP^5 to a smooth line γof the projective plane C^1 P^2, bears two plane singular lines, and finally the submanifold X^3, corresponding in RP^5 to a smooth line γof the projective plane C^0 P^2, bears one singular line.