I am puzzled by the Picard group of $X=\text{Proj }k[x_0,x_1,x_2,x_3,x_4]/(x_0x_1-x_2x_3)$. I can calculate its Weil's divisor class group ($\mathbb{Z}\oplus \mathbb{Z}$) because it is the projective cone of $\text{Proj }k[x_0,x_1,x_2,x_3]/(x_0x_1-x_2x_3)$. And the relevant propositions have been analyzed in the second chapter of GTM 52. However, due to the singularity at $[0,0,0,0,1]$, I cannot conclude that Picard group is the Weil's class group. I once tried to use Cech's cohomology to calculate $H^1(X,\mathcal{O}_X^*)$. But I don't think that I can handle the difficulty in concrete calculation.
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1GTM 52 is Hartshorne's "Algebraic Geometry", I believe. – diracdeltafunk Dec 25 '22 at 18:13