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?



Can a singular Deligne-Mumford stack have a smooth coarse space?
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 **A**2, 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 **A**1, which is smooth.


