Let $S$ be a smooth projective surface over $\mathbb{C}$ and $L$ be an ample line bundle on $S$. Let $d$ be a positive integer such that $dL$ is very ample and $D$ a very general member of the linear system $|dL|$. Let $T_D \rightarrow S$ be the degree $d$ cyclic cover branched along $D$. Assume that the Picard number $ \rho(S) =1 $.

Q.(Edited) Is the picard number $\rho(T_D)$ $1$ for a sufficiently large integer $d$?

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    $\begingroup$ Not in general. In fact, the pullback of $\textrm{Pic}(S)$ inside $\textrm{Pic}(T)$ consists of those divisor classes which are invariant under the $\mathbb{Z}/d \,\mathbb{Z}$-action. $\endgroup$ – Francesco Polizzi Jul 19 '13 at 19:38
  • $\begingroup$ Thank you for your comment. I edited the question a bit. If you have some specific example, please let me know. Thanks. $\endgroup$ – tarosano Jul 19 '13 at 22:45
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    $\begingroup$ For double covers, see: Buium, Alexandru Sur le nombre de Picard des revêtements doubles des surfaces algébriques. CRAS 296 (1983), no. 8, 361–364. $\endgroup$ – Felipe Voloch Jul 19 '13 at 23:35

No. Let $S=\mathbb{P}^2$, let $L=\mathcal{O}(1)$, and let $d=2$. The double cover of $\mathbb{P}^2$ branched along a smooth quadric is isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$.

  • $\begingroup$ Thank you very much for the answer. I should consider more carefully. $\endgroup$ – tarosano Jul 19 '13 at 22:29
  • $\begingroup$ I forgot that the usual Noether-Lefschetz theorem requires that the degree of hypersurfaces is $\ge 4$. I edited the question. If you can still find a counter example, please let me know. Thanks. $\endgroup$ – tarosano Jul 19 '13 at 22:44

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