A Severi type theorem for surfaces in P^6

Let X⊂P^N be a projective, non-degenerate, irreducible smooth variety of dimensionn. After giving the definition ofgeneralised 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.