A severi type theorem for surfaces in p⁶

Let X subset of P-N be a projective, non-degenerate, irreducible smooth variety of dimension n. After giving the definition of generalised OADP-variety (one apparent double point), i.e. varieties X such that: n(k + 1) - (N - r)(k - r) + r = N, there is one apparent (k + 1)-secant (r- 1)-space to a generic projection of X from a point, we concentrate in studying generalised OADP-surfaces in low dimensional projective spaces, and the main result of this paper is the classification of smooth surfaces in P-6 with one 4-secant plane through the general point of P-6.