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Given a (smooth)Fano 3-fold $X$, Sokurov proved that the fundamental linear system contains a smooth surface. My question is :

If the Picard number of X is 1,Is there such a smooth surface(in the fundamental linear system) also with Picard number 1?.

P.S. if we work on picard number 1 Fano manifold, then $Pic(X)$ is generated by an ample bundle $H$, by fundamental linear system of $X$ I mean $|H|$.

Thx.

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  • $\begingroup$ What do you mean by "the fundamental linear system"? $\endgroup$ – Sasha Nov 4 '13 at 15:37
  • $\begingroup$ In case of index 2 your question does not make much sense for the surfaces in $|H|$, which are Del Pezzo surfaces hence have Picard number $>1$. On the other hand it makes sense for the anticanonical system, see my answer below. $\endgroup$ – abx Nov 5 '13 at 9:19
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This is true if the anticanonical system (which you call the fundamental system) is very ample. Indeed one can take a Lefschetz pencil and apply Theorem 1.4 of SGA 7, Exposé XIX (Noether's theorem by Deligne). Now we know the complete list of Fano threefolds with Picard number 1 , thanks to Iskovskikh; there are only 2 or 3 cases where $-K$ is not very ample, maybe they can be checked by hand.

Afterthought : Actually the result does not hold for at least one family of Fano threefolds with Picard number 1, namely those for which the anti canonical map $\pi:X\rightarrow Q\subset {\Bbb P}^4$ is a double covering of a smooth quadric in ${\Bbb P}^4$. Indeed the smooth surfaces in $\ |-K_X|\ $ are of the form $S_h:=\pi^{-1}(Q_h)$, where $Q_h$ is a smooth hyperplane section of $Q$. Since $\mathrm {Pic}(Q_h)=\Bbb{Z}^2$, the Picard number of $S_h$ is $\geq 2$.

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