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Modulo the definition, I’ve created Picard scheme. One thing I couldn’t tell, is there a standard term in nlab for the “fiber category” of a stack? I mean if $F:C\to D$ fibers $C$ over $D$ then if you pick some object $X$ from $D$ the category $C_X$ consisting of objects that go to $X$ and morphisms that go to $id_X$.
is there a standard term in nlab for the “fiber category” of a stack?
I don’t think there is currently. I would call this simply “the value of the stack at $X$”. But feel free to introduce an entry for this, if you need it.
If you’re thinking of a stack as a pseudofunctor, then “the value of the stack at $X$” seems most sensible. If you prefer to think of it as a fibration, then I think “the fiber over $X$” also makes perfect sense. Of course in neither case does it depend on the pseudofunctor/fibration being a stack.
Yes, that’s why I clarified by merely saying “if $F:C\to D$ fibers $C$ over $D$” to point out that if the term existed it probably wasn’t just for stacks.
The Idea-section at Picard scheme claimed (,aybe that was me, I forget) that it is the geometric incarnation of the Picard group. I have added the warning that often (usually) it is just the connected component of that which is being referred to. In fact I have added the following paragraph:
Often one considers just the connected component $Pic_X^0$ of the neutral element in $Pic_X$, and often (such as in the discussion below, beware) it is that connected component (only) which is referred to by “Picard scheme”. The quotient $Pic_X/Pic_X^0$ is called the Néron-Severi group of $X$.
Related to this: the discussion on the page as of now is not very transparent conceptually. I have added links to Akhil Mathew’s nice posts on this, which go a bit further in actually explaining what’s going on. But it seems to me this could further be improved on.
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