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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 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 $k+1$.

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!

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    $\begingroup$ Typo: your surface $S$ is in $\Bbb{P}^{n+2}$. $\endgroup$ – abx Oct 9 '18 at 4:29
  • $\begingroup$ Perhaps you need to choose the degrees $d_i$ so that $I_{C_1\cap\cdots\cap C_k}(d_i)$ is globally generated or something like that? $\endgroup$ – Jonathan Frink Oct 9 '18 at 8:20
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    $\begingroup$ @Jonathan Frink: The $C_i$ are assumed to be disjoint. $\endgroup$ – abx Oct 9 '18 at 9:56
  • $\begingroup$ Sorry, that is supposed to be $I_{C_1\cup \cdots \cup C_k}(d_i)$.. $\endgroup$ – Jonathan Frink Oct 9 '18 at 12:36
  • $\begingroup$ Variant 1: No, in my counter-example you can actually take any smooth 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
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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.

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