# Can a singular Deligne-Mumford stack have a smooth coarse space?

Let XX be a Deligne-Mumford stack and let XX \to X be a coarse moduli space. Suppose that X is smooth. Is XX smooth? If not, what is an example? What if XX is of finite type over C (the complex numbers)? What are conditions we can put on XX to make this true?

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What does it mean for XX \to X to be a coarse moduli space? –  Kevin H. Lin Oct 21 '09 at 14:21
The map XX --> X is a coarse space if: 1) It is universal for maps to algebraic spaces. 2) It induces a bijection on geometric points (so kbar points for algebraically closed fields kbar). Examples: the coarse space of M_1,1 is A^1, given by the j-invariant map (this is actually a little hard to prove). Easier: the coarse space of BG is a point (or for BG over S, S). This one follows directly from the definitions. Quotients by finite groups are also easy to work out. –  David Zureick-Brown Nov 2 '09 at 18:03
Links: Anton's notes (section 38 on Keel-Mori): math.berkeley.edu/~anton/written/Stacks/Stacks.pdf. Conrad: notes on Keel-Mori: math.stanford.edu/~conrad/papers/coarsespace.pdf. Alper: math.columbia.edu/~jarod/stacks_guide.pdf has a lot good pointers. –  David Zureick-Brown Nov 2 '09 at 18:05
One more link: the appendix of Kai-Wei's thesis has some good stuff in it too: math.princeton.edu/~klan/academic.html –  David Zureick-Brown Nov 2 '09 at 19:21

The answer is yes, a singular DM stack can have a smooth coarse space. Let U=Spec(k[x,y]/(xy)) be the union of the axes in A2, and consider the action of G=Z/2 given by switching the axes: x→y and y→x. Then take XX to be the stack quotient [U/G]. This is a singular Deligne-Mumford stack (since it has an etale cover by something singular), but its coarse space is A1, which is smooth.

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what's the difference between this stack quotient and the quotient of the affine line (char is not 2) by x --> -x? –  shenghao Jan 3 '10 at 21:16
The quotient of the line by that action is smooth (since it has an etale cover by something smooth, namely the affine line), but this stack is not smooth (since it has an etale cover by something non-smooth, namely the axes). –  Anton Geraschenko Jan 3 '10 at 22:25