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stjc
stjc
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Given a (smooth)Fano 3-fold X$X$, 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.

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?.

Thx.

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.

Source Link
stjc
stjc
  • 1.1k
  • 1
  • 8
  • 12

Picard number of fundamental divisor of Fano 3-fold

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?.

Thx.