It is "well-known" that the stack of polarized varieties is an algebraic stack with quasi-compact and separated diagonal.


In particular, if $(X,L)$ and $(Y,M)$ are polarized schemes over a scheme $S$, the Isom-scheme $$Isom((X,L),(Y,M))$$ is separated and quasi-compact over $S$.


I had the feeling that these Isom-schemes should be of finite type and affine over $S$ and surely this is well-known. In other words,


Is the diagonal of the stack of polarized varieties of finite type?


Is it affine?


I did not find an answer in the stacks project. I'm currently looking at work of Rydh and others, but haven't gotten lucky yet.



Are automorphism groups of polarized varieties of finite type
Breaking with my habit of writing everything in comments, this is Section 2.1 of my article with de Jong.


MR2745688 (2012e:14073) Reviewed   

Starr, Jason(1-SUNYS); de Jong, Johan(1-CLMB)   

Almost proper GIT-stacks and discriminant avoidance. (English summary)   
 
Doc. Math. 15 (2010), 957–972.   

14J10 (14L15)   

<http://www.math.uiuc.edu/documenta/vol-15/29.pdf>


