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.



Stack with affine stabilizers but not quasi-affine diagonal
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**).


