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.
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 $\geq 2$.
