# Hyperplane sections of surfaces in $\mathbb{P}^3$

Suppose that $$X \subset \mathbb{P}^3$$ is a smooth complex projective surface of degree $$d \geq 4$$. Then the Noether-Lefschetz theorem states that if $$X$$ is very general, then $$\mathrm{Pic}(\mathbb{P}^3) \rightarrow \mathrm{Pic}(X)$$ is an isomorphism, so in particular,

$$\mathrm{Pic}(X) \cong \mathbb{Z} \cdot \mathcal{O}_X(1).$$

Of course, there are special surfaces for which this fails and the Picard number jumps. However, I was wondering: are there any special surfaces $$X$$ for which $$\mathcal{O}_X(1)$$ is not primitive in $$\mathrm{Pic}(X)$$? In other words, can there exist a line bundle $$L$$ on $$X$$ and a positive integer $$r \geq 2$$ such that $$L^{\otimes r} \cong \mathcal{O}_X(1)$$? Examples of this or references to a proof of the contrary would be appreciated.

This cannot happen. Grothendieck proved that for any such surface, the natural map $$\operatorname{Pic} \mathbb{P}^3\to \operatorname{Pic} \hat{X}$$ is an isomorphism where $$\hat{X}$$ is the formal completion. So it suffices to prove that the cokernel of the natural map $$\operatorname{Pic}\hat{X}\to \operatorname{Pic} X$$ is torsion free. If $$X _n$$ denotes the $$n^ {th}$$ order thickening defined by the ideal $$I^n$$ where $$I$$ defines $$X$$, we have a natural exact sequence, $$0\to I^n/I^{n+1}\to \mathcal{O}_{X_{n+1}}^*\to\mathcal{O}_{X_n}^*\to 0$$. Taking cohomologies, we see that the desired cokernel is filtered by subspaces of $$H^2(I^n/I^{n+1})$$ and since we are over complex numbers, these are torsion free.