Tame Artin Stacks -- Angelo Vistoli

I am reporting on joint work with Dan Abramovich and Martin Olsson.

A Deligne-Mumford stack F is tame when the order of the automorphism group of an object of F(k), where k is an algebraically closed field, is prime to the characteristic of k. Tame stacks behave better than wild ones in several ways; in particular, there is a good notion of stable map into a tame Deligne-Mumford stack.

We introduce a more general notion of tame Artin stack, give a complete local description of tame Artin stacks, and show that stable maps into a tame Artin stack have the same good properties as in the Deligne-Mumford case. As a particular case we obtain a stack of ramified mu_n-coverings that is flat and proper over Z.

Also, an offshoot, we characterize the finite group schemes (over an arbitrary base) that are linearly reductive.