On multiosculating spaces

The aim of this paper is to study varieties with second Gauss map not birat ional. In particular we classify such varieties in dimension two. We prove that there are two types of surfaces S of P-n(C), with n > 5, not satisfyin g Laplace equations, with second Gauss map t(2) not birational: i) surfaces such that the image of the second Gauss map is one-dimensional and containing a one-dimensional family of curves. Each curve of the family is contained in some P-3 subset of or equal to P-n. ii) surfaces such that the second Gauss map is generically finite or degree at least two. In this case the image of the second Gauss map is two-dimens ional, locally embedded in a Laplace congruence and meeting the general gen eratrix in more than one point.