All Questions
40 questions
7
votes
1
answer
482
views
A "comprehensive" family of abelian varieties
I'm looking for a family of abelian varieties $A\rightarrow S$ over a base that is finite type over $\mathbb{Q}$ (or $\mathbb{Z}$) that is "comprehensive" in the following sense: for every ...
- 333
1
vote
0
answers
120
views
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 ...
- 31
2
votes
1
answer
383
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 \...
- 45
1
vote
1
answer
291
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 $...
- 8,998
3
votes
0
answers
198
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 ...
- 12.1k
1
vote
1
answer
271
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}_{...
user168627
0
votes
0
answers
411
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 $\...
- 297
3
votes
1
answer
427
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 ...
- 445
1
vote
1
answer
232
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{\...
- 31
3
votes
1
answer
262
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 ...
- 560
2
votes
1
answer
197
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 ...
- 1,593
4
votes
1
answer
287
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 ...
- 8,998
4
votes
1
answer
270
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 ...
user117617
1
vote
1
answer
221
views
Compactifications of spaces of morphisms
Let us denote by $Mor_3(\mathbb{P}^1,\mathbb{P}^3)$ the spaces of degree three morphisms $f:\mathbb{P}^1\rightarrow\mathbb{P}^3$,
$$f(x_0,x_1)=[f_0(x_0,x_1):f_1(x_0,x_1):f_2(x_0,x_1):f_3(x_0,x_1)]$$
...
user117617
5
votes
0
answers
347
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-...
- 556
3
votes
1
answer
264
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 ...
user97096
4
votes
1
answer
355
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,...
- 325
1
vote
0
answers
189
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 ...
user85435
2
votes
0
answers
159
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 ...
- 21
5
votes
1
answer
524
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 ...
- 18.7k
4
votes
0
answers
180
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 ...
- 1,509
2
votes
0
answers
272
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 ...
- 101
2
votes
1
answer
223
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 ...
user68440
1
vote
1
answer
757
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 ...
user56259
3
votes
1
answer
675
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 $\...
user61586
6
votes
1
answer
568
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 ...
user58018
1
vote
1
answer
327
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 ...
user58604
2
votes
1
answer
363
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 ...
user55899
9
votes
1
answer
937
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 ...
- 8,998
4
votes
2
answers
451
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 $m$, ...
- 4,492
6
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 ...
- 73
1
vote
1
answer
257
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 ...
- 8,998
4
votes
0
answers
478
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 ...
- 203
6
votes
3
answers
834
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 $[...
- 3,779
4
votes
0
answers
486
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 ...
- 3,779
4
votes
0
answers
289
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 ...
- 381
5
votes
0
answers
239
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 ...
- 381
12
votes
1
answer
2k
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}\...
- 10.7k
27
votes
6
answers
4k
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 ...
- 281
12
votes
0
answers
1k
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{...
- 3,284