Resumen de Rectas proyectivas sobre R_álgebras bidimensionales y cuádricas en el espacio proyectivo

Santiago Mazuelas Franco

• The classic Riemann Sphere construction, that is, the real plane compactification by means of a sphere or equivalently via the complex projective line; is widely known and used. This construction achieves, for example, a clear description of cycles and a linear representation of the inversive group of the euclidean plane by means of the Moebius group. In this paper we study the analog construction for the other types of bidimensional ]_algebras: paracomplex and dual numbers, and for the other types of real and irreducible quadrics in the projective space: hyperboloid and real cone. We make a especial emphasis in the definition of the projective lines over rings with zero divisors. Thereby we show that it is possible to perform a construction com-pletely analogous to the Riemann Sphere for the other types of bidimen-sional algebras and quadrics in the projective space and we also show the relation of these constructions with the hyperbolic and degenerate metrics of the real plane.