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Give an example of a stack X with affine stabilizer groups and separated but not quasi-affine diagonal.

Remarks:

1) If X has finite stabilizer groups then the diagonal is quasi-finite and separated, hence quasi-affine (Zariski's MT).

2) If we drop the condition that the diagonal of X is separated, it is easy to find examples.

3) The stabilizer groups of X are affine if and only if they are quasi-affine.

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1 Answer

Here is an example:

In X13 of "Faisceaux amples sur les schemas en groupes", Raynaud provides an example of a group scheme G -> S in chacteristic 2 where

  1. S is a local regular scheme of dimension 2.
  2. G -> S is smooth, separated and quasi-compact.
  3. The fibers of G ->S are affine and the generic fiber is connected.

such that G -> S is not quasi-projective.

Therefore, the classifying stack BG has affine stabilizers but does not have a quasi-affine diagonal.

On the other hand, in VII 2.2, Raynaud proves that if G -> S is a smooth, finitely presented group scheme such that

  1. S is normal.
  2. G -> S has connected fibers.
  3. The maximal fibers are affine.

then G -> S is quasi-affine.

Question: Is the above statement true if (2) is weakened to require that the number of connected components over a fiber s \in S be prime the characteristic of the residue field k(s)?

Of course, one would really like to know if the statement is true if G->S is not necessarily flat so that one could apply it to the inertia stack.

On a related note, Raynaud also provides an example in VII3 of a smooth quasi-affine group scheme G -> A^2 over a field k with connected fibers but which is not affine. The classifying stack BG gives an example of stack with affine and connected stabilizers but with non-affine inertia stack. In the example of a scheme with non-affine diagonal, the inertia is of course affine. It's also easy to provide examples of non-affine group schemes with affine but non-connected fibers (eg. the group scheme obtained by removing the non-identity element over the origin from Z/2Z -> A^2).

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