Questions tagged [coarse-moduli-spaces]
The coarse-moduli-spaces tag has no usage guidance.
23
questions with no upvoted or accepted answers
12
votes
0answers
931 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{...
7
votes
0answers
181 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 ...
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
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 ...
5
votes
0answers
123 views
global units on moduli spaces of abelian varieties
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 relative dimension $g,$ ...
4
votes
0answers
134 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 ...
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 ...
4
votes
0answers
380 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 ...
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 ...
3
votes
0answers
146 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 ...
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\}...
3
votes
0answers
397 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 ...
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 ...
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 $\...
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 ...
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}...
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 ...
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 ...
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 ...
2
votes
0answers
657 views
Coarse Moduli space of proper Deligne-Mumford stacks
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?
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 ...
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 ...
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 $\...
