2
$\begingroup$

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 $d$?

$\endgroup$
  • 2
    $\begingroup$ 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
  • 3
    $\begingroup$ 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
4
$\begingroup$

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 $\mathbb{P}^1\times \mathbb{P}^1$.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.