On real complete intersections

Let X be a real algebraic subvariety of P-R(n). We prove that if n is large enough with respect to the degrees of the equations defining X, these degrees being odd the variety of real r-planes contained in X is not empty and has the expected dimension. This variety can be disconnected, and the r-planes it parametrizes may cover only a proper semi-algebraic subset of X with a non-empty interior. However, through a general point of X passes a real conic contained in X. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.