Inertia jumps again

In this post I want to continue the discussion of the previous post by asking: How do space and automorphisms get mixed up in codimension 1.

Everybody’s favorite example of this phenomenon is the algebraic stack [A^1/mu_n] over a field. Namely this is a smooth separated stack of dimension 1 with generically trivial stabilizer and special stabilizer the group scheme mu_n of nth roots of 1. Consider the morphism

[A^1/mu_n] —> A^1

given by z maps to z^n on the covering A^1 of the stack. This is an isomorphism everywhere except over 0 where we get as stack theoretic fiber the algebraic stack [Spec(k[z]/(z^n)/mu_n]. One of the many cute things about this example is that if you look at the canonical morphism

[Spec(k[z]/(z^n)/mu_n] —> [Spec(k)/mu_n]

then the push forward of the structure sheaf corresponds to the regular representation of mu_n. I suggest we compare this with the fact that the push forward of the structure sheaf via the morphism Spec(k) —> [Spec(k)/mu_n] corresponds to the regular representation as well. For me this signifies that [Spec(k[z]/(z^n)/mu_n] is really a “single point”. Another fact is that if you consider the inverse of the morphism above, namely

A^1 – {0} —> [A^1/mu_n]

then the corresponding mu_n-torsor over A^1 – {0} is a generator of H^1_{fppf}(A^1 – {0}, mu_n). There are more canonical and coordinate independent ways of formulating these properties, which we leave to the reader…

Now I think that for any smooth separated algebraic stack over a field of characteristic zero having generically trivial stabilizer this is the only kind of jump that happens in codimension 1. (Haven’t proved it. If you add the condition that the stack is Deligne-Mumford then this is easier to prove.) In characteristic p > 0 there are many other finite groups that can occur as jumps in codimension 1. This is true for example because large finite p-groups act faithfully on k[[t]] if k is a field of characteristic p; the simplest action being perhaps the action of Z/pZ given by t maps to t/(1 + t). Note: Z/pZ is very different from mu_p in characteristic p.

If we look at still smooth but not necessarily separated algebraic stacks (back in characteristic zero) then many other jumps of automorphism groups happen in codimension 1. Here are some examples:

  1. The stack [A^1/G_m] gives an example where G_1 = {1} and G_0 = G_m.
  2. The stack [symmetric bilinar forms/GL_n] gives an example where G_1 = O(n) and G_0 is an extension of G_m x O(n – 1) by an n-1 dimensional additive group.
  3. The stack [skew symmetric bilinear forms/GL_{2n}] gives an example where G_1 = Sp(2n) and G_0 is an extension of GL_2 x Sp(2n-2) by an 2(2n – 2) dimensional additive group.
  4. The stack M_1 of genus 1 curves gives an example where G_1 is an elliptic curve semidirect Z/2Z and G_0 is an elliptic curve semi-direct Z/6Z.
  5. The stack \bar M_1 of generalized genus 1 curves gives an example where G_1 is an elliptic curve semidirect Z/2Z and G_0 is G_m semidirect Z/2.

There is much worse that can happen (namely, nonseparated group schemes) if you allow the diagonal morphism to be nonseparated itself. But somehow if the stack is smooth, the characteristic is zero, and the diagonal morphism is separated, then I think (this is nonsense see below) that the picture should always be that in codimension 1 the stack fibers over [A^1/G_m] or [A^1/mu_n] with “fibre” B(H) where H is a flat group scheme. The proof should be that one takes H the closure of the generic stabilzer and then one divides it out.

[Edit: Jarod Alper pointed out that the last paragraph I also have to allow for [P^1/G_a] action via translation locally around infinity as a possibility. Maybe there are even others? Answer: yes, many. Will explain in next post.]