Let $F$ be the stack (over some base scheme $S$) which associates to every scheme $X$ the groupoid of invertible sheaves on $X$. Then by the general theory (see for example Aoki's paper) $F$ is algebraic. My question is: Can you write down a nice presentation $P \to F$ explicitly, i.e. a surjective smooth morphism from a scheme $P$?

Also, is there a reference for the observation that $\mathsf{Qcoh}(F)$ is equivalent to the category of $\mathds{Z}$-graded quasi-coherent modules on $S$?


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