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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→D fibers C over D then if you pick some object X from D the category CX consisting of objects that go to X and morphisms that go to idX.
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→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 Pic0X of the neutral element in PicX, and often (such as in the discussion below, beware) it is that connected component (only) which is referred to by “Picard scheme”. The quotient PicX/Pic0X 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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