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I know the Picard group of a smooth two dimensional quadric surface is $\mathbb Z^2$, but I am wondering if the computation can be generalized to higher dimension? In particular, is the Picard group of a $n$-dimensional smooth quadric hypersurface $\{\sum_{i=0}^{n+1}x_i^2=0\}\subset\mathbb P^{n+1}$ known?

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    $\begingroup$ Smooth hypersurfaces in projective spaces $\mathbb{P}^n$ with $n\geq 4$ has Picard group $\mathbb{Z}$ generated by the hyperplane section. $\endgroup$
    – Mohan
    Commented Apr 18, 2023 at 2:12
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    $\begingroup$ .... this usually goes by the name "Lefschetz hyperplane section theorem". You can read about it in the book Lazarsfeld - Positivity in Algebraic Geometry I. $\endgroup$
    – Daniel Loughran
    Commented Apr 18, 2023 at 10:32
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    $\begingroup$ This also follows by very simple methods as well: the blowing up of a smooth quadric at a point equals the blowing up of projective space along a smooth quadric whose dimension is two less. $\endgroup$
    – Jason Starr
    Commented Apr 18, 2023 at 11:12
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    $\begingroup$ The nice comment of Jason, (project the quadric from a point of itself down one dimension), also explains why the theorem fails precisely for surfaces: smooth quadrics are reducible precisely in dimension zero! $\endgroup$
    – roy smith
    Commented Apr 18, 2023 at 19:18

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