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 …

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 …

Consider the stack $Bun_{r,d}^{ss}$ of rank $r$ semi-stable vector bundle of degree $d$ over a fixed curve. There exists also a coarse moduli space $M$ built via GIT. Over the stab …

Hi! I am stuck with the following question: suppose we have a semisimple connected algebraic group acting on a quasi-affine variety X by closed orbits, and suppose that inertia i …

Let $M_g$ be the moduli space of smooth projective geometrically connected curves over a field $k$ with $g\geq 2$. Note that $M_g$ is not complete. Does $M_g$ contain an elliptic …

This question is somehow a more generalized re-edit of a former question of mine: http://mathoverflow.net/questions/119339/glueing-flat-families-of-objects-over-a-blow-up I guess …

Non-hyperelliptic curves of genus three are smooth quartics. Is the moduli space of such curves affine? I think this follows from a more general result on smooth complete intersec …

Let $M_h$ be the moduli space of canonically polarized varieties with Hilbert polynomial $h$. Let $M_h \to A_g$ be the Albanese map, with $g$ an integer which depends on $h$ and $A …

So, when people say, "the moduli problem of classifying elliptic curves over $\mathbb{C}$ with level $N$ structure", there are usually two associated functors I've seen: $P_N : \ …

It took me some effort to work out Gerashenko's nice simple example http://mathoverflow.net/questions/1565/can-a-singular-deligne-mumford-stack-have-a-smooth-coarse-space/1584#1584 …

Let $M_{g} \to \mathbb{Z}$ be the coarse moduli scheme of non-singular genus g curves over $\mathbb{Z}$. That is, suppose that $M_{g}$ co-represents the functor $M^{\sharp}_{g} : \ …

Degeneration of certain functions such as theta functions or Green's functions in the moduli space $\overline{\mathcal{M}_g}$ of stable curves of genus $g$ has been studied quite a …

Hi, Is the modular curve defined as the quotient of the upper half-plane by an arithmetic group $ \Gamma $ always a moduli space of elliptic curves with extra structure? I know t …

Let F be a finite type proper Deligne-Mumford Stack over a perfect field. Is it true that the coarse moduli space of F is proper?

This is a question from a colleague. Let $A_{g,d,n}$ be the coarse moduli space over $\mathbb Z$ (of the moduli stack, or assume $n\ge3$ if you like) of abelian schemes of relati …