Timeline for Can a singular Deligne-Mumford stack have a smooth coarse space?
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5 events
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| Jan 4, 2010 at 23:54 | comment | added | David Zureick-Brown | (No guarantee on that last sentence). | |
| Jan 4, 2010 at 23:53 | comment | added | David Zureick-Brown | Ah, so in that case the fibers are gerbes (and I think the total space is a gerbe). In general, the coarse space does not represent the functor you wrote down; it is an intermediate step in proving that a coarse space exists, but you may still need too modify that functor more. Even for M_1,1 I believe the coarse space differs from this functor. For instance, two families of curves with the same `j-invariant' give the same map to M_1,1, but there isn't necessarily a flat cover of the base where they become isomorphic (though there is on some open subset of the base). | |
| Jan 4, 2010 at 0:43 | comment | added | shenghao | I seem to remember this from the book <champs algebrique> by LMB. 1. In the book, a coarse moduli space of a stack XX is defined to be the functor X sending U to the set of isom classes in XX(U). And XX->X is a gerbe. When XX is DM over k, I guess (without checking details) these two notions agree. 2. At least for any field-valued pt x:Spec k -> X, the fiber f^{-1}(x) is a fppf-gerbe over x (fppf-locally there exists an obj over x, and any two objects are isomorphic over \bar{k}, hence isom over some finite ext'n of k), and therefore smooth. Does this imply that f is smooth? | |
| Jan 3, 2010 at 23:07 | comment | added | David Zureick-Brown | Why is XX --> X a gerbe? | |
| Jan 3, 2010 at 21:22 | history | answered | shenghao | CC BY-SA 2.5 |