# Are automorphism groups of polarized varieties of finite type

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.

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

• Section 2.1 gives a proof for why the Isom-scheme is affine. Whether the Isom-scheme is of finite type over the base is not discussed, as far as I can tell. Jul 31, 2015 at 15:04
• I suppose that we do not explicitly say that it is finite type over the base, but we factor the morphism as (first) a closed immersion (automatically finite type) followed by an explicit finite type affine morphism. The $\mathcal{O}_U$-modules $f_*\mathcal{N}$ and $g_*\mathcal{L}$ are locally free sheaves of finite ranks $r$ and $s$. Thus $\text{Isom}_U(f_*\mathcal{N},g_*\mathcal{L})$ is empty if $r\neq s$, and it is (Zariski locally) $\text{GL}_{r,U}$ if $r$ equal $s$. Jul 31, 2015 at 15:18
• Yes, I realized this shortly after writing my comment. :) I guess one could also say that you are showing the Isom-schemes map finitely to the automorphism group of some projective bundle over $U$ associated to some vector bundle on U, right? Jul 31, 2015 at 15:24
• Basically that is what we were saying. Also Section 2.1 is basically a summary of what is in other references (which we list there). We spelled things out a bit more than other authors. However, as you identified, we also left many things unsaid. Jul 31, 2015 at 15:27
• Also, slightly related to $\text{Isom}$ begin finitely generated is Lemma 4.7 in another of my papers with de Jong, "Every Rationally Connected Variety ...". Jul 31, 2015 at 15:32