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nLab > Latest Changes: Picard scheme

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    • CommentRowNumber1.
    • CommentAuthorhilbertthm90
    • CommentTimeSep 13th 2011

    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→DF:C\to D fibers CC over DD then if you pick some object XX from DD the category C XC_X consisting of objects that go to XX and morphisms that go to id Xid_X.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 13th 2011

    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 XX”. But feel free to introduce an entry for this, if you need it.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeSep 13th 2011

    If you’re thinking of a stack as a pseudofunctor, then “the value of the stack at XX” seems most sensible. If you prefer to think of it as a fibration, then I think “the fiber over XX” also makes perfect sense. Of course in neither case does it depend on the pseudofunctor/fibration being a stack.

    • CommentRowNumber4.
    • CommentAuthorhilbertthm90
    • CommentTimeSep 13th 2011

    Yes, that’s why I clarified by merely saying “if F:C→DF:C\to D fibers CC over DD” to point out that if the term existed it probably wasn’t just for stacks.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2014
    • (edited May 14th 2014)

    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 0Pic_X^0 of the neutral element in Pic XPic_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 0Pic_X/Pic_X^0 is called the Néron-Severi group of XX.

    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.

  6. Think I fixed a mathematical typo

    Yongyi

    diff, v20, current

  7. Think I fixed a mathematical typo

    Yongyi

    diff, v20, current

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