I would like to know whether the following kind of Noether--Lefschetz statement is true, and if so, to have a reference.

Fix natural numbers $d_1,\ldots,d_n$ such that $\sum_i d_i \geq n+3$. Denote by $d$ the product $\prod_i d_i$. Let $C_1, \ldots C_k$ be smooth disjoint curves in $\mathbf P^{n+2}$, with $\operatorname{deg}(C_i) <d$ for each $i$.

Then a

very generalcomplete intersection surface $S \subset \mathbf P^{n+2}$ of multidegree $(d_1\ldots,d_n)$ containing all the curves $C_i$ has Picard number $k+1$.

Of course many restrictions and generalisations of the setup are possible, so please feel free to modify conditions as you wish.

**Update:** As the good answer of abx shows, the statement I ask for cannot be true. Let me therefore ask about a couple of variants:

**Variant 1:** with the setup as above, do the classes of the $C_i$ together with the class of a hyperplane section of $S$ generate $\operatorname{Pic}(S)$, or at least a finite-index subgroup?

**Variant 2:** Is either the original statement or Variant 1 true if we further require that the $C_i$ be smooth rational curves?

It would be great to know any positive statements along these lines!

anysmooth quartic surface containing a line, so the Picard number may be any number between 2 and 20. Variant 2 makes sense, but I doubt that the answer is known. $\endgroup$ – abx Oct 10 '18 at 4:16