Factoriality and Néron-Severi groups

Let $X\subset \Ps^{2m+1}$ be a projective variety with isolated singularities, complete intersection of a smooth hypersurface of degree $k$, with a smooth hypersurface $F$ of degree $n>k$. Denote by $NS_m(F)$ and $NS_m(X)$ the $m$-th N\'eron-Severi groups. We prove that if $rkNS_m(F)=1$ then $rkNS_m(X)=1$. Moreover we prove that if $F\in\mid\ic_{X,\Ps^{2m+1}}(n)\mid$ is general and $n>max\{k, 2m+1\}$, then the natural map $NS_m(X)\otimes\bQ \to NS_m(F)\otimes \bQ$ is surjective. When $X$ is a threefold we deduce that $X$ is factorial if and only if $rkNS_2(F)=1$. This allows us to prove the existence of factorial threefolds with many singularities.