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.


1 Answer 1


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)

  • $\begingroup$ 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. $\endgroup$
    – Pancho
    Jul 31, 2015 at 15:04
  • $\begingroup$ 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 $s$. $\endgroup$ Jul 31, 2015 at 15:18
  • $\begingroup$ 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? $\endgroup$
    – Pancho
    Jul 31, 2015 at 15:24
  • $\begingroup$ 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. $\endgroup$ Jul 31, 2015 at 15:27
  • $\begingroup$ 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 ...". $\endgroup$ Jul 31, 2015 at 15:32

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