Let $\mathcal{X}$ be an Artin stack. In "Abramovich, Graber, Vistoli - Twisted bundles and admissible covers", the authors describe a procedure to rigidify $\mathcal{X}$ by a centr …

This is a construction [Definition 6.1] given in the paper D-equivalence and K-equivalence by Kawamata. Let $X$ be a normal quasiprojective variety such that the canonical …

Let M be your favorite moduli stack over the field of complex numbers. Is it reasonable to expect M to be a Deligne-Mumford stack? I know this is true for the moduli space of cur …

Suppose I have a certain (contravariant) moduli functor $M:Schemes \to Groupoids$ that is represented by a quotient stack $[X//G]$ where $X$ is a scheme and $G$ a linearly reductiv …

Suppose that we have two Grothendieck sites, their associated sheaves $\mathcal{E}=\rm{Sh}(\bf{C},J)$ and $\mathcal{F}=\rm{Sh}(\bf{D},K)$ and a geometric surjection $f:\mathcal{E}\ …

Suppose I have an Artin stack $\mathfrak M$ locally of finite-type over $\mathbb C$ with presentation $M\rightarrow \mathfrak M$. Suppose further that $\mathfrak M$ "represents" ( …

It is well known that a one-dimensional smooth Deligne-Mumford stack (over $\mathbb{C}$) could be described as a collection of its "stacky" points (finitely many) on its coarse mod …

Let $F: (Sch)^{o}\to (Set)$ be a functor that admits a coarse moduli $Y$ (a scheme). We can consider $Y$ as a representable functor $h_{Y}: (Sch)^{o}\to (Set)$. Is there a direct …

I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphis …

Well, the title almost says it all. I would like to list as many examples as possible of moduli functors, for which a coarse moduli space does not exist (and maybe explain why). So …

I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization. The Statement Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a …

It is well known and more or less proven in Hartshorne's 'Algebraic Geometry' (p. 209) that for every noetherian scheme $X$ and every collection of abelian sheaves $\mathcal{F}_i$ …

1) Is it true that any Deligne-Mumford stack is locally a quotient stack $[X/G]$ with a finite group $G$? 2) Is it true that any Deligne-Mumford stack can be globally presented as …

Consider a stack $\mathcal{X}$ over $\mathbb{C}$ as a category fibred in groupoids over the category of schemes. Let $\mathcal{X}^s$ be the $\pi_0$ of this category, i.e. objects o …

Let $X$ be something. (smooth and projective variety over C are my assumptions) The stack $M$ parameterising coherent sheaves on $X$ splits as a disjoint union of open and closed s …