Questions tagged [coarse-moduli-spaces]
The coarse-moduli-spaces tag has no usage guidance.
53
questions
4
votes
1answer
131 views
Smoothness of moduli spaces of stable maps
If $X$ is a projective variety the moduli space of stable maps $\overline{M}_{0,0}(X,\beta)$ is a normal variety with finite quotient singularities.
Can the pairs $(X,\beta)$ such that $\overline{M}_{...
2
votes
1answer
142 views
Is the set of hyperelliptic curves with a K-point closed?
I am actually interested in the same question for more general kinds of curves, but I will be specific.
Let $K$ be a field and $\overline{K}$ be an algebraic closure of $K$. Let's say that a "...
0
votes
0answers
176 views
Definition of Hitchin map
This may be a dumb question.
$\mathcal{M}(r,d)$ is a coarse moduli scheme for semistable pairs $(E,\phi:E \rightarrow K_X \otimes E)$ of rank $r$, degree $d$ on a smooth projective curve $X$ over $\...
4
votes
0answers
131 views
Family over the coarse moduli space of curves
Let $k$ be an algebraically closed field. As the coarse moduli space of curves $M_g$ of genus $g$ over $k$ is not a fine moduli space, it does not have a universal family. But I am wondering if it has ...
3
votes
0answers
138 views
The coarse moduli space of a weighted projective line
Fix two positive integers $a$ and $b$. Consider a weighted projective line $\mathbb{P}(a,b)$ as a quotient stack
$$[(\mathbb{C}^2-\{0\})/\mathbb{C}^*]$$
where $\mathbb{C}^*$ acts on $\mathbb{C}^2-\{0\}...
1
vote
1answer
136 views
Pullback of boundary divisors under forgetful maps
Let $\overline{\mathbf{M}}_{0,n}$ be the moduli space of stable $n-$pointed smooth rational curve of genus zero and $\overline{\mathbf{U}}_{0,n}$ the universal family described by $\pi_n:\overline{\...
3
votes
1answer
241 views
$\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties
Reading M. Hindry and J. H. Silverman (Diophantine Geometry-An Introduction), I find the claim that $\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties. Mumford ...
2
votes
0answers
79 views
The space $M_g$ with the complex structure induced from $T_g$ is a coarse moduli space for compact Riemann surfaces of genus $g$
Proposition: The space $M_g$ with the complex structure induced from $T_g$ is a coarse moduli space for compact Riemann surfaces of genus $g$.
In the proof of (1), I wonder why the holomorphic map $\...
3
votes
2answers
463 views
When is the coarse moduli space of genus $g$ stable curves smooth?
Let $\overline{\mathcal{M}}_g$ be the moduli stack of genus $g$ stable (nodal) curves and let $\overline{M}_g$ denote its coarse moduli space. In 1969, in the paper "The irreducibility of the space of ...
5
votes
2answers
394 views
Fundamental group of $M_g^\circ$
Let's work over the complex numbers $\mathbb{C}$. Let $g\geq3$ be an integer. Let $\mathcal{M}_g$ be moduli stack of smooth genus $g$ curves. Let $M_g$ be the corresponding coarse moduli scheme. They ...
7
votes
2answers
409 views
Understand the difference between two stacks
Let us work over $\mathbb{C}$. Let $G$ be a finite group, acting on $\mathbb{A}^1$ via a character, and let $H$ be the kernel of the action.
Assume that $\mathbb{A}^1$ is the coarse moduli space of ...
2
votes
1answer
142 views
When is the moduli of generalized parabolic bundles with fixed determinant smooth?
Let $X$ be a smooth, projective curve of genus at least $2$, $x_1, x_2$ two distinct closed points, $d$ an odd integer and $\alpha$ a positive real number less than $1$. By a generalized parabolic ...
5
votes
1answer
220 views
Intermediate moduli spaces of stable maps
In the following paper:
A. Mustata, M. A. Mustata, "Intermediate moduli spaces of stable maps", Invent. math. 167, 47–90 (2007)
the authors introduced a variation on moduli spaces of stable maps ...
2
votes
0answers
119 views
Flip of moduli space of stable maps
Let $\overline{M}_{0,2}(G(3,7),4)$ be the moduli space of $2$-pointed degree $4$ stable maps to the Grassmannian of $3$-planes in $\mathbb{P}^7$. Consider the divisor $\Delta$ whose general point is a ...
5
votes
1answer
239 views
Blowing-up projective spaces of parametrized rational curves
Consider the projective space $\mathbb{P}^N$ parametrizing morphisms $f:\mathbb{P}^1\rightarrow\mathbb{P}^n$, $f(x,y) = [f_0(x,y):\dots:f_n(x,y)]$ of degree $d$.
Let $Z_i\subset\mathbb{P}^N$ be the ...
2
votes
1answer
186 views
Compactifications of spaces of morphisms
Let us denote by $Mor_3(\mathbb{P}^1,\mathbb{P}^3)$ the spaces of degree three morphisms $f:\mathbb{P}^1\rightarrow\mathbb{P}^3$,
$$f(x_0,x_1)=[f_0(x_0,x_1):f_1(x_0,x_1):f_2(x_0,x_1):f_3(x_0,x_1)]$$
...
5
votes
0answers
233 views
Good quotients and coarse moduli spaces
I started study the moduli space theory, by Newstead's book ''Lectures on moduli problems and orbit spaces". Theorem 3.21 says that given a variety $X$ and a line bundle $L$ over $X$, then for any L-...
5
votes
1answer
204 views
Linear systems on moduli spaces of stable maps
I am studying the general theory of moduli spaces of stable maps, in particuar of the moduli spaces $\overline{M}_{0,n}(\mathbb{P}^r,d)$ of degree $d$ stable maps from a rational curve with $n$ marked ...
4
votes
1answer
223 views
“Generalized” clutching maps between moduli spaces of curves
Let $P=\{1,\dots,n\}$ and $S\subseteq P$. The map $$\nu:\overline{\mathcal{M}}_{i,S\cup\{q\}}\to \overline{\mathcal{M}}_{g,P},$$ which attaches to a curve in the domain a pointed genus $g-i$ curve $[D,...
2
votes
0answers
91 views
Reference request: Families of curves and associated mapping classes
In Teichmüller theory, we consider families of genus $g$ smooth complex projective curves with $n$ distinguished points.
Assume $2g-2+n>0$ and, for convenience, $(g,n)\neq(1,1),(2,0)$.
Denote ${T}...
1
vote
0answers
177 views
Mumford's claim on the quasi-projectiveness of the coarse moduli of ppav over $\mathbb{Z}$
In paragraph 3 of chapter 7 of Mumford's Geometric Invariant Theory, the author proves that the coarse moduli space $A_{g,d,n}$ of abelian varieties of dimension $g$, with a degree $d^2$ polarization ...
2
votes
0answers
155 views
Are these moduli problems of curves “well-behaved”?
Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$.
Let $H_{X,d}$ be the Hilbert scheme ...
5
votes
1answer
388 views
Essential dimension and the moduli space of abelian varieties
The following problem is listed here: http://www-personal.umich.edu/~erman/Papers/Questions2.pdf and attributed to Vistoli:
Let $\mathcal A_g$ denote the moduli stack of principally polarized abelian ...
4
votes
1answer
206 views
Non canonical singularities of moduli spaces of curves
Is it true that for any $g\geq 1$ and $n$ such that $\overline{M}_{g,n}$ has dimension at least two the locus in $\overline{M}_{g,n}$ parametrizing reducible curves which are union of an elliptic ...
4
votes
0answers
149 views
When does a “universal” quot scheme exist?
Suppose $M$ is a moduli space of semistable sheaves on a projective variety $X$. Let $v$ be some the discrete invariants. I would like to form a space $\mathcal Q(v) \rightarrow M$, where the fiber ...
1
vote
0answers
90 views
critiria to prove that a morphism is an embedding
I have a forgetful map between moduli spaces, I want to prove that it is an embedding, In fact, I have a reductive algebric group (which is not constant) over a curve $X$ whose geniric fiber is ...
2
votes
0answers
173 views
local universal sheaf (moduli of stable sheaves)
I do not know much about moduli of sheaves and I wanted to shows that for a smooth (projective) family over a discrete valuation ring of mixed characteristics (relative dimension 3), the locally free ...
5
votes
1answer
521 views
Quadrics and Moduli Spaces
It is well known that $\overline{M}_{0,5}$, the moduli space of $5$-pointed rational curves, can be realized as the blow-up of $\mathbb{P}^2$ in four general points. Therefore, we may interpret $\...
3
votes
1answer
518 views
Rational curves in projective spaces
Let $X\subset(\mathbb{P}^{N})^n$ be the variety defined as follows: $(p_1,...,p_n)\in (\mathbb{P}^{N})^n$ such that there exists a rational curve $C$ of degree $d$ with $p_1,...,p_n\in C$.
Is there a ...
7
votes
0answers
180 views
Groupoid cardinality of DM stack and point counting on coarse moduli spaces
Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.)
Under which ...
9
votes
1answer
471 views
Rationality of moduli spaces of rational curves
Let $\overline{M}_{0,n}$ be the moduli space of Deligne-Mumford stable pointed rational curves, and let us consider the quotient $\widetilde{M}_{0,n} = \overline{M}_{0,n}/S_n$.
Clearly, there is a ...
4
votes
1answer
264 views
Canonical bundle of moduli space of rational curves and automorphisms
Let $\overline{M}_{0,n}$ be the usual Deligne-Mumford compactification of $M_{0,n}$ the moduli space of smooth $n$-pointed rational curves.
The canonical divisor $K_{\overline{M}_{0,n}}$ can be ...
2
votes
1answer
303 views
Moduli of curves in characteristic zero
Let $K$ be a field of characteristic zero, and let $\overline{K}$ be its algebraic closure. Let $\overline{M}_{g,n}(K)$ and $\overline{M}_{g,n}(\overline{K})$ be the coarse moduli spaces parametrizing ...
1
vote
0answers
192 views
Coarse moduli spaces and rational points [closed]
Let $K$ be a field (not necessarily algraically closed). Let $\mathcal{F}$ be a contravariant functor from the category of schemes over $K$ to sets and $M$ be a coase moduli space for the functor. So, ...
9
votes
1answer
644 views
Singularities of moduli spaces of curves
Let $\overline{M}_{g,n}$ be the moduli space of $n$-pointed genus $g$ Deligne-Mumford stable curves. This is a normal projective scheme. Then
$$codim_{\overline{M}_{g,n}}Sing(\overline{M}_{g,n})\geq ...
4
votes
2answers
327 views
A question on the existence of the quotient of the Hilbert scheme of tricanonical curves
In order to construct the coarse moduli scheme of smooth projective curves of genus $g$, the classical results of Mumford (using the numerical criterion of stability) say that for large enough $m$, ...
5
votes
1answer
1k views
${\rm Ext}^1$ and extensions of line bundles on a curve
I am confused about the following. I know that for two line bundles $L_1, L_2$ on an algebraic curve $C$ the vector space ${\rm Ext}^1(L_1,L_2)$ classifies isomorphism classes of rank two vector ...
1
vote
1answer
218 views
Moduli spaces admitting birational morphisms over moduli spaces of curves
There are many alternative compactifications of $M_{g,n}$ which live naturally under the classical Deligne-Mumford compactification $\overline{M}_{g,n}$.
For instace the moduli spaces of weighted ...
4
votes
0answers
379 views
Points of moduli space of semistable sheaves and S-equivalence classes
Let $X$ be smooth projective and connected curve over an algebraically closed field $k$. One knows the description of $k$-valued points of the moduli space $M_X$ of semistable vector bundles of fixed ...
6
votes
3answers
651 views
examples of moduli functors for which coarse moduli space does not exists
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, examples such as $[...
3
votes
0answers
396 views
Stack of vector bundles (on a curve) over a strictly semi-stable point of the moduli space
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 stable locus of $M$ it ...
8
votes
1answer
1k views
On the coarse moduli space of a stack
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 of $\mathcal{X}^s$ ...
3
votes
0answers
70 views
Are two “nice” transformation groupoids with the same coarse moduli and isomorphic inertia isomorphic?
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 is flat.
Suppose we ...
10
votes
6answers
3k 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 ...
4
votes
0answers
265 views
Does the Albanese map satisfy Torelli's theorem
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 $A_g$ the moduli space ...
5
votes
0answers
230 views
Is the moduli space of genus three smooth quartics affine?
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 intersections, but I'm ...
12
votes
1answer
1k views
what exactly is the moduli functor for classifying elliptic curves with (full) level N structure?
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 : \textbf{Ell}\...
26
votes
6answers
3k views
Does the moduli space of smooth curves of genus g contain an elliptic curve
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 curve?
The answer ...
2
votes
0answers
266 views
Is there an algebraic analogue of the degeneration of riemann surfaces in M_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 alot. The idea is to ...
12
votes
0answers
930 views
What is $M_g$ over a finite field, really?
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} : \text{Sch} \to \text{...
