Skip to main content
MathOverflow

Return to Question

added 2 characters in body
Source Link
JKDASF
JKDASF
  • 231
  • 1
  • 5

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 hyperplanehypersurface $\{\sum_{i=0}^{n+1}x_i^2=0\}\subset\mathbb P^{n+1}$ known?

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 hyperplane $\{\sum_{i=0}^{n+1}x_i^2=0\}\subset\mathbb P^{n+1}$ known?

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?

Source Link
JKDASF
JKDASF
  • 231
  • 1
  • 5

Computing Picard groups of arbitrary quadric hyperplane

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 hyperplane $\{\sum_{i=0}^{n+1}x_i^2=0\}\subset\mathbb P^{n+1}$ known?