Search Results

Tame Artin stacks (in the sense of Abramovich, Olsson and Vistoli, https://math.berkeley.edu/~molsson/tame.pdf) with quasi-projective moduli spaces will have property 2: the line bundle is the pullback … As to property 1, a line bundle will not be enough, but you can get a version for quotient tame Artin stacks using a generating sheaf (in the sense of Olsson and Starr, https://math.berkeley.edu/~molsson …

The stack of curves of genus 0 with at most one node is a quotient stack (see The integral Chow ring of the stack of at most 1-nodal rational curves, but Edidin and Fulghesu), so you are fine in this …

In the smooth case, I think that the answer is positive over an arbitrary regular excellent base. The argument was in my PhD thesis; it was done over a field, but I think it adapts to this case. Cove …

It is certainly not true that $\mathcal X \to \mathcal X^H$ is a good moduli morphism, unless $H$ is linearly reductive, because when you push forward the cohomology of $H$ will come into play. On th …

Consider the fiber product $S \times_{\frak M} M$; its projection onto $S$ is smooth and surjective, hence open. Since $S$ is quasi-compact, there exists a quasi-compact open subset $U$ of $S \times_{ …

When you post a question, it would be good if you include enough explanations not to force the interested reader to go search for a paper online. Anyway, the general question is: suppose that we have …

From the definition it follows easily that tame stacks with affine moduli spaces have the property you require. … There are several different characterizations of tame stacks; see the paper "Tame stacks in positive characteristic" by Dan Abramovich, Martin Olsson and myself. …

By a family of gerbes you mean, I suppose, a gerbe over $X \times \mathbb A^{1}$. In any case, it has a class in $\mathrm H^0(\mathbb A^1, \mathrm R^2 \mathrm{pr}_{2*}\mathbb G_{\rm m})$. Since $\math …

If $X \to Y$ is a proper morphism of DM stacks, where $X$ has finite inertia (the hypotheses in my paper are more stringent, but the theory has been refined since then), there exists a finite map $V \to …

For the examples you mention, this boils down to representability of Hom sheaves of flat finitely presented proper schemes, which is due to Grothendieck. These Hom sheaves are not of finite type, thou …

As to why algebraic stacks are always assumed to be stacks in groupoids, there are several things I could say, but the honest answer is that I don't know the deep reason for this. … not stacks in groupoids. …

I have heard this result attributed to Kai Behrend, who apparently came up with it while writing his part of the now defunct many-authored stack book. However it is certainly possible that someone els …

I think that what they might have in mind is that for non-quasi-separated Deligne-Mumford algebraic stacks one should not assume that the diagonal is represented by schemes, but by algebraic spaces. … For quasi-separated Deligne-Mumford stacks this implies representability by schemes, but this is not true in general. …

They only work with stacks that are locally finitely presented. …

I don't think this is true. Let $X$ be the quotient of the action of $\mathbb Q$ on $\mathbb R$ by translation. This is a sheaf, and its automorphism groups are trivial. Suppose that there exist a loc …