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Feb 7, 2016 at 11:01 comment added Lennart Meier For me, a weighted projective (stacky) line is the stack quotient quotient $(Spec K[x,y]-0)/\mathbb{G}_m$, where $\mathbb{G}_m$ acts by $\lambda^a$ on $x$ and by $\lambda^b$ on $y$ for some weights $a,b\in\mathbb{Z}_{>0}$. We are in the world of the linked paper if $a$ and $b$ are coprime (no generic stabilizer). Now note that $\mathbb{Z}w\oplus \mathbb{Z}z/(aw = bz) \cong \mathbb{Z}$ with generator $cw+dz$, where $cb+ad = 1$. Thus, in the intersection of the two concepts, we get the same answer for $Pic$.
Feb 5, 2016 at 18:45 comment added pbelmans Is there a difference in terminology for weighted projective line or Picard group that explains why you say the Picard group is $\mathbb{Z}$ whereas e.g. section 2 of arxiv.org/abs/1409.7050v1 says that is a bit bigger than that?
Feb 1, 2016 at 18:17 history answered Lennart Meier CC BY-SA 3.0