ON MULTIOSCULATING SPACES

The aim of this paper is to study varieties with second Gauss map not birational. 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 satisfying 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 ? P ⁿ .

ii. surfaces such that the second Gauss map is generically finite of degree at least two. In this case the image of the second Gauss map is two-dimensional, locally embedded in a Laplace congruence and meeting the general generatrix in more than one point.