Conditions for "bootstrapping" a smooth DM stack? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:51:54Z http://mathoverflow.net/feeds/question/19195 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19195/conditions-for-bootstrapping-a-smooth-dm-stack Conditions for "bootstrapping" a smooth DM stack? Johan 2010-03-24T14:28:26Z 2010-03-24T16:42:07Z <p>In the preprint "<a href="http://arxiv.org/abs/0708.1254" rel="nofollow">Smooth toric DM stacks</a>", 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:</p> <p>1) start with a scheme (the coarse moduli scheme) with at worst finite quotient singularities, and take the associated canonical stack;</p> <p>2) use a root stack construction to possibly add some extra stack structure to divisors (given by an integer for each divisor);</p> <p>3) finally add a gerbe.</p> <p>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.</p> <p>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)?</p>