Questions tagged [coarse-moduli-spaces]

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Geometric quotients of DM stacks by group actions

Let $G$ be a finite group acting on a DM stack $X$ and $Y$ the quotient stack. I.e., $Y \to BG$ has fiber $X$. Is there a geometric quotient $X/G$ in some sense? I want automorphism groups of points ...
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Local description of the universal family $\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}$

I would like to get an understanding of the notion of geometric fibers of the universal family: $$\pi: \overline{\mathbf{U}}_{0,n} \longrightarrow \overline{\mathbf{M}}_{0,n}.$$ In fact Knudsen show ...
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2 votes
1 answer
206 views

$G$-invariant morphism and coarse moduli spaces

Let $G$ be an algebraic group acting on $X$ (a finite type scheme on $k$). A $G$-invariant $k$-morphism $f : X \rightarrow S$ is a map such that the following commute: $\require{AMScd}$ \begin{CD} G \...
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The weight of a weighted filtration is given (for large $m$) by a polynomial

Let $I$ be an homogeneous ideal of $k[x_0, \dots, x_n]$. Suppose to give integral weights $\lambda_0, \dots, \lambda_n$ to $x_0, \dots, x_n$. We assign a weight to every homogeneous polynomial of ...
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233 views

Tangent space to spaces of maps

Let $B = \{x_1,\dots,x_{d-2},y_1,\dots,y_k\}$ be a subscheme of $d-2+k$ distinct points of $\mathbb{P}^1$, and $g:B\rightarrow \mathbb{P}^2$ be a morphism mapping $x_1,\dots,x_{d-2}$ to a fixed point $...
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3 votes
0 answers
181 views

Interesting stacks with affine space as coarse moduli

I am looking for examples of Deligne-Mumford stacks whose coarse moduli space is $\mathbb{A}^n$ or at least an open subscheme of $\mathbb{A}^n$ whose complement has codimension $2$. (Thus the whole ...
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1 vote
1 answer
167 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}_{...
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3 votes
1 answer
167 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 "...
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0 votes
0 answers
246 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 $\...
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3 votes
1 answer
214 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 ...
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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\}...
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1 answer
165 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{\...
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3 votes
1 answer
252 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 ...
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82 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 $\...
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2 answers
630 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 ...
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7 votes
2 answers
554 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 ...
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7 votes
2 answers
421 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 ...
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2 votes
1 answer
165 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 ...
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5 votes
1 answer
244 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 ...
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1 answer
245 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 ...
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1 vote
1 answer
188 views

Compactifications of spaces of morphisms

Let us denote by $Mor_3(\mathbb{P}^1,\mathbb{P}^3)$ the spaces of degree three morphisms $f:\mathbb{P}^1\rightarrow\mathbb{P}^3$, $$f(x_0,x_1)=[f_0(x_0,x_1):f_1(x_0,x_1):f_2(x_0,x_1):f_3(x_0,x_1)]$$ ...
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5 votes
0 answers
261 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-...
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3 votes
1 answer
216 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 ...
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4 votes
1 answer
260 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,...
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2 votes
0 answers
92 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}...
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1 vote
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179 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 ...
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2 votes
0 answers
158 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 ...
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5 votes
1 answer
436 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 ...
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4 votes
0 answers
154 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 ...
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  • 1,471
1 vote
0 answers
92 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 ...
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  • 197
2 votes
0 answers
201 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 ...
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2 votes
1 answer
213 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 ...
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1 vote
1 answer
635 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 ...
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3 votes
1 answer
583 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 $\...
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7 votes
0 answers
189 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 ...
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6 votes
1 answer
516 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 ...
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1 vote
1 answer
277 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 ...
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2 votes
1 answer
330 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 ...
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1 vote
0 answers
202 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, ...
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9 votes
1 answer
717 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 ...
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4 votes
2 answers
386 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 $m$, ...
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5 votes
1 answer
2k 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 ...
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1 vote
1 answer
228 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 ...
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4 votes
0 answers
417 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 ...
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6 votes
3 answers
703 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 $[...
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  • 3,665
3 votes
0 answers
412 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 ...
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  • 3,665
8 votes
1 answer
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 $\mathcal{X}^s$ ...
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3 votes
0 answers
71 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 ...
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4 votes
0 answers
270 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 ...
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  • 381
5 votes
0 answers
235 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 ...
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