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 $i$.
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> Then a *very general* complete intersection surface $S \subset \mathbf P^{n+2}$ of multidegree $(d\_1\ldots,d\_n)$ containing all the curves $C\_i$ has Picard number $k+1$.
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> 


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!



Conditional Noether--Lefschetz theorems
I think this is false. Consider a very general quartic surface $S\subset\mathbb{P}^3$ containing a line $\ell$. Projecting from $\ell$ defines an elliptic fibration $S\rightarrow \mathbb{P}^1$. Choose some fibers $C\_1,\ldots ,C\_k$, with $k\geq 6$. For degree reasons $S$ is the only quartic surface containing all the $C\_i$, but its Picard number is 2, while $k$ can be arbitrarily large.


Here the $C\_i$ have degree 3; I must say I do not know the answer if the $C\_i$ are lines or conics.


