  For a projective variety $Z$ and for any integer $p$, define the $p$-th
N\'eron-Severi group $NS_p(Z)$ of $Z$ as the image of the cycle map
$A_{p}(Z)\to H_{2p}(Z; \mathbb{C})$. Now let $X\subset \Ps^{2m+1}$ ($m\geq 1$)
be a projective variety of dimension $2m-1$, with isolated singularities,
complete intersection of a smooth hypersurface of degree $k$, with a
hypersurface of degree $n>max\{k, 2m+1\}$, and let $F$ be a general
hypersurface of degree $n$ containing $X$. We prove that the natural map
$NS_m(X)\to NS_m(F)$ is surjective, and that if $dim NS_m(F)=1$ then $dim
NS_m(X)=1$. In particular $dim NS_m(X)=1$ if and only if $dim NS_m(F)=1$. When
$X$ is a threefold (i.e. $m=2$) we deduce a new characterization for the
factoriality of $X$, i.e. that $X$ is factorial if and only if $dim NS_2(F)=1$.
This allows us to give examples of factorial threefolds, in some case with many
singularities. During the proof of the announced results, we show that the
quotient of the middle cohomology of $F$ by the cycle classes coming from $X$
is irreducible under the monodromy action induced by the hypersurfaces of
degree $n$ containing $X$. As consequences we deduce a Noether-Lefschetz
Theorem for a projective complete intersection with isolated singularities,
and, also using a recent result on codimension two Hodge conjecture, in the
case $X\subset \Ps^{5}$ is a threefold as before, we deduce that the general
hypersurface $F$ of degree $n$ containing $X$ verifies Hodge conjecture.
