Questions tagged [stacks]
In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.
513 questions
2
votes
1
answer
354
views
k-th Chow Group and k-th graded part of K_0 ismorphic for DM-stacks?
If X is an algebraic scheme, K_0(X) has a filtration by taking the subgroups generated by coherent sheaves whose support as at most dimension k. The associated graded groups are the quotients, and ...
- 3,917
13
votes
1
answer
655
views
Examples of algebraic stacks without coarse moduli space?
Keel-Mori's theorem says an algebraic stack with a finite diagonal over a scheme S has a coarse moduli space. What is an example of an algebraic stack without coarse moduli space?
- 215
6
votes
3
answers
2k
views
What are Log Stacks
So, I've been running in both stacky circles and logarithmic circles and I've been wondering: is there a definition of log stack that is "useful"? I can imagine two such definitions:
1) A log stack ...
- 16k
65
votes
17
answers
17k
views
Good introductory references on algebraic stacks?
Are there any good introductory texts on algebraic stacks?
I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also ...
- 1,538
11
votes
6
answers
4k
views
What are some examples of coarse moduli spaces?
It took me some effort to work out Gerashenko's nice simple example Can a singular Deligne-Mumford stack have a smooth coarse space? of a DM stack non-equisingular with its coarse moduli space, which ...
14
votes
6
answers
2k
views
Does every morphism BG-->BH come from a homomorphism G-->H?
Given a homomorphism f:G→H between smooth algebraic groups, we get an induced homomorphism of algebraic stacks Bf:BG→BH, given by sending a G-torsor P over a scheme X to the H-torsor PxGH, ...
9
votes
1
answer
2k
views
Kodaira-Spencer Theory and moduli of curves
I was looking at a paper of Farkas and the following confusing point came up.
Let $\mathscr{M}_g$ be the moduli stack of smooth genus $g$ curves and let $\pi: \mathscr{C} \to \mathscr{M}_g$ be the ...
- 10.5k
14
votes
2
answers
2k
views
Can a singular Deligne-Mumford stack have a smooth coarse space?
Let XX be a Deligne-Mumford stack and let XX \to X be a coarse moduli space. Suppose that X is smooth. Is XX smooth? If not, what is an example? What if XX is of finite type over C (the complex ...
- 10.5k
17
votes
2
answers
3k
views
Are curves with `fractional points' uniquely determined by their residual gerbes?
One makes precise the vague notion of "curve with a fractional point removed" (see for instance these slides) using stacks -- one should really consider Deligne-Mumford stacks whose coarse spaces are ...
- 10.5k
25
votes
4
answers
2k
views
algebraic group G vs. algebraic stack BG
I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...
11
votes
2
answers
2k
views
Finiteness conditions on simplicial sheaves/presheaves
Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
- 5,637
9
votes
1
answer
1k
views
Stack with affine stabilizers but not quasi-affine diagonal
Give an example of a stack X with affine stabilizer groups and separated but not quasi-affine diagonal.
Remarks:
1) If X has finite stabilizer groups then the diagonal is quasi-finite and separated, ...
- 5,039
3
votes
2
answers
857
views
Is there an example of an algebraic stack whose closed points have affine stabilizers but whose diagonal is not affine?
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 ...
- 10.5k