One makes precise the vague notion of "curve with a fractional point removed" (see for instance [these slides](http://www-math.mit.edu/~poonen/slides/campana_s.pdf)) using stacks -- one should really consider Deligne-Mumford stacks whose coarse spaces are curves, and the "fractional points" correspond to the residual gerbes at the stacky points.


One example: let a,b,c > 1 be coprime integers and let S be the affine surface given by the equation x^a + y^b + z^c = 0. Then there is a weighted Gm action on S (t sends (x,y,z) to (t^bc x, t^ac y, t^ab z) and one can check that the stack quotient [S-{0}/Gm] has coarse space P^1 and, that since the action is free away from xyz = 0 but has stabilizers at those 3 points, one gets a stacky curve with three non-trivial residual gerbes.



> 
> **Question**: Are two 1-dimensional DM stacks with isomorphic coarse spaces 
>  and residual gerbes themselves isomorphic?
> 
> 
> 


I have an idea for how to prove this when the coarse space is P^1, but in general don't know what I expect the answer to be. Also, one may have to restrict to the case when the coarse space is a smooth curve.


This might be analogous to the statement that the `angle' of a node of a rational nodal curve with one node doesn't affect the isomorphism class of the curve.


Also, the recent papers of Abromivich, Olsson, and Vistoli (on stacky GW theory) may be relevant.



Are curves with `fractional points' uniquely determined by their residual gerbes?
Here's an example of two non-isomorphic Deligne-Mumford stacks whose coarse spaces are **A**1, and the only non-trivial residual gerbe in each case is B(**Z**/2) at the origin.


First, take the **Z**/2 action on **A**1 given by reflection around 0, given by x→-x. The stack quotient [**A**1/(**Z**/2)] has coarse space **A**1 and there's a B(**Z**/2) gerbe at the origin. Note that this stack is smooth since it has an etale cover by something smooth.


On the other hand, you can take the stack I defined in [this answer](https://mathoverflow.net/questions/1565/can-a-singular-deligne-mumford-stack-have-a-smooth-coarse-space/1584#1584): the stack quotient of the coordinate axes in **A**2 by the **Z**/2 action which switches the two axes. The coarse space is again **A**1 and the stack has a B(**Z**/2) gerbe at the origin. Note that this stack is *not* smooth since it has an etale cover by something singular, so it cannot be isomorphic to the previous stack.


An example where both stacks are smooth
---------------------------------------


The first stack will be the same [**A**1/(**Z**/2)] I used above.


Let G be the affine line with a doubled origin, regarded as a group over **A**1 (most of the fibers are trivial groups, but the fiber over the origin is **Z**/2). This G has the trivial action on **A**1. Consider the quotient stack [**A**1/G]=B**A**1 G. This is a DM stack whose coarse space is **A**1, and there's a B(**Z**/2) gerbe at the origin. Note that this stack is smooth since it has an etale cover by a smooth scheme (namely, **A**1). Note that this stack has non-separated diagonal (since the pullback G→**A**1 is non-separated), but the diagonal of [**A**1/(**Z**/2)] is separated, so the two stacks are non-isomorphic.


