  We work over an algebraically closed field of arbitrary characteristic.
  Ellingsrud-Peskine proved that smooth surfaces in P^4 are subject to strong
limitations. Their whole argument is derived from the fact that the sectional
genus of surfaces of degree d lying on a hypersurface of degree s varies in an
interval of length \frac{d(s-1)^2}{2s}. The aim of the present paper is to show
that for smooth codimension two subvarieties of P^n, n\geq 5, one can get a
similar result with an interval whose length depends only on s. The main point
is Lemma 1.1 whose proof is a direct application of the positivity of N_X(-1)
(where N_X is the normal bundle of X in P^n). We get a series of (n-3)
inequalities; the first one of which being in the paper of Ellingsrud-Peskine,
the second was obtained in a preliminary version (arXiv:math.AG/0406497) by an
essentially equivalent but more geometric argument. Then we first derive two
consequences: 1) roughly speaking, (Thm. 2.1, Remark 2.2) the family of
"biliaison classes" of smooth subvarieties of P^5 lying on a hypersurface of
degree s is limited; 2) the family of smooth codimension two subvarieties of
P^6 lying on a hypersurface of degree s is limited (Thm. 1.4). The result
quoted in 1) is not effective, but 2) is. In the last section we try to obtain
precise inequalities connecting the usual numerical invariants of a smooth
subcanonical subvariety X of P^n, n\geq 5 (the degree d, the speciality index
e, the least degree, s, of an hypersurface containing X). In particular we
prove (Thm. 3.12): s\geq n+1.
