Burt Totaro has a result that for a certain class of algebraic stacks, having affine diagonal is equivalent to the stabilizers at closed points begin affine. Is there an example of this equivalence failing in general?



Is there an example of an algebraic stack whose closed points have affine stabilizers but whose diagonal is not affine?
The answer is yes since a fiberwise condition (such as affine stabilizers) does not imply a global condition (such as affine diagonal) without extra hypotheses (such as having the resolution property). Think of quasi-finite+proper <=> finite.


There are (non-separated) *schemes* with non-affine diagonal, for example, two copies of the affine plane glued together outside the origin.


Also see the related [question](http://www.mathoverflow.net/questions/76/).


