# Conditional Noether--Lefschetz theorems

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 $$\operatorname{deg}(C_i) for each $$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!

• Typo: your surface $S$ is in $\Bbb{P}^{n+2}$. – abx Oct 9 '18 at 4:29
• 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? – Jonathan Frink Oct 9 '18 at 8:20
• @Jonathan Frink: The $C_i$ are assumed to be disjoint. – abx Oct 9 '18 at 9:56
• Sorry, that is supposed to be $I_{C_1\cup \cdots \cup C_k}(d_i)$.. – Jonathan Frink Oct 9 '18 at 12:36
• 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. – abx Oct 10 '18 at 4:16

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.