Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

In the preprint "Smooth toric DM stacks", Fantechi, Mann and Nironi define the stacks of their title, and show that each of these can be obtained through the following sequence of steps:

1) start with a scheme (the coarse moduli scheme) with at worst finite quotient singularities, and take the associated canonical stack;

2) use a root stack construction to possibly add some extra stack structure to divisors (given by an integer for each divisor);

3) finally add a gerbe.

Not all smooth DM stacks can be obtained this way, e.g. for $n>3$ take the global quotient $\mathbb{C}^n/S_n$, where the symmetric group $S_n$ acts by permuting the factors of $\mathbb{C}^n$. The coarse moduli scheme is smooth here, and there is no gerbe, but the stack doesn't seem to arise as a root stack.

Are there conditions known for reasonable (finite type over a field,...) smooth DM stacks under which the stack can be obtained by the "bootstrapping" procedure described above (or a similar one)?

share|improve this question
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.