Questions tagged [moduli-space]
A Moduli space is a space in algebraic geometry whose points are geometric objects or isomorphism classes of these kinds of objects.
197
questions
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When is $\operatorname{Spec} A^G \cong (\operatorname{Spec} A)/G$ true?
Let $G$ be a group acting on a ring $A$. I would like to know in which generality we know that $\operatorname{Spec} A^G \cong (\operatorname{Spec} A)/G$. Moreover, when this is true, it also holds for ...
6
votes
2answers
210 views
What does it mean for points of the modular curve $X(N)$ to be “defined over $\mathbb{F}_p$”?
I'm trying to study a collection of elliptic curves over some fixed finite field $\mathbb{F}_p$. By browsing the literature and discussing with my supervisor, it seems like it will be fruitful to ...
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0answers
22 views
Isomorphism of curves induces isomorphism of moduli spaces of vector bundles on the curves?
Suppose we have an isomorphism of projective curves $f : C \to C'$. Let $M_C(\mu)$ denote the moduli space of slope stable vector bundles on $C$ with slope $\mu$. Do we get an induced isomorphism $F : ...
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0answers
18 views
Historical significance of moduli spaces
Last semester I took a class on symplectic topology where we proved Gromov's theorem using moduli spaces of spheres. This semester, I am taking algebraic geometry following Hartshorne and after ...
1
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1answer
98 views
Quotient of $GL_n$ by unipotent group and maximal torus
Assume we are working over a field $k$ and $G= GL_n$. Then we get the (standard) maximal torus $T$, the unipotent group $U$ and the Borel $B$ of upper triangular matrices. More generally we have we ...
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0answers
61 views
Stable graph of stable curves
Let $\Gamma = (\mathbf{V},\mathbf{H}, \mathbf{L},g, v,i,\ell)$ be a stable graph.
Let $(\mathcal{C},p_1,\cdots,p_n)$ be a stable curve with the dual curve $\Gamma$ and let $(\tilde{\mathcal{C}}_v,(q_h)...
6
votes
1answer
224 views
What is the definition of moduli space, in math vs in physics?
It is easy to find that there are many questions regarding moduli space on MSE: https://math.stackexchange.com/search?q=what+is+moduli+space
But it seems to me that this phrase, moduli space, may mean ...
2
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0answers
169 views
Applications of the Chinese remainder Theorem to the study of the Hilbert scheme of points and $(\mathfrak{m},l)$-squeezed ideals.
The following construction gives a relation between the Chinese Remainder Theorem (CRT), the Noether nomalization lemma (NNL) and cofinite ideals in finitely generated $k$-algebras.
Let $k$ be any ...
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0answers
51 views
Construction of Hilbert Variety (in Harris' Algebraic Geometry)
My question intends to take up a detail in the construction of the Hilbert variety in Harris' book
Algebraic Geometry; A First Course (p 274).
Let $S=K[X_0,X_1,..., X_n]$ be the ring of polynomials in ...
0
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0answers
40 views
Hyperplane $\Phi_X$ in Chow construction (Algebraic Geometry by Joe Harris)
I'm trying to figure out what Harris wanted to say in following
construction, called Chow construction (Algebraic Geometry by Joe Harris, p. 269):
The first construction of the parameter space for ...
-6
votes
2answers
259 views
Some integers related to the Hilbert scheme of points in the plane. [closed]
This question is related to another question posed on this site.
Let me recall the construction: Let $A:=k[x,y]/I$ with $k$ the complex numbers (or any algebraically closed field) and $\dim_k(A)< \...
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0answers
40 views
Reduction of structure group $GL_n$ of the endomorphism bundle to the centraliser.
I was reading this proof from http://www.numdam.org/item?id=AST_1982__96__1_0 where of structure group of the bundle $EndE$ which is taken to be $GL_n$ is reduced to the centraliser of $GL_n$. The ...
1
vote
1answer
170 views
Can we factor this quotient ring.
Let $I \subset \mathbb{C}[x,y]$ be an ideal such that $\dim \mathbb{C}[x,y]/I =n$ for some natural number $n$. For any point $p \in \mathbb{C}^2$ we define $I_p=(x-p_x,y-p_y)^n+I$. I want to prove ...
3
votes
1answer
66 views
Coarse Moduli space of plane cubics
I am studying Joe Harris' Algebraic Geometry: A First Course, the section
on Moduli Spaces, pg 278. I am stuck in a subtle point.
Harris gives on p 279 an argument why there is no coarse moduli space
...
1
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0answers
28 views
Question about the proof that there does not exist a fine moduli space for endomorphisms of $n$-dimensional vector spaces.
I am reading Mumford's 1970 Oslo lecture on moduli theory.
In this lecture, he defines a family of n-dimensional vector space endomorphisms over a scheme $T$ to be a pair $(\mathcal{E}, \phi)$ where $\...
5
votes
1answer
50 views
Help with moduli spaces of four-marked spheres
I'm thinking about the moduli space of the four-punctured sphere where some of the removed points are distinguishable and some are indistinguishable. I believe there should be some covering maps ...
0
votes
1answer
76 views
Isomorphism between $\overline{U}_{0,4}$ and the degree $5$ Del Pezzo Surface
Tacitely, I am working over the field of complex numbers!
Let $\overline{M}_{0,4}\cong\mathbb{P}^1$ be the compactification of the moduli space of $4$-pointed stable rational curves. The relevant ...
7
votes
1answer
144 views
What is Representation of Surface Groups?
I had a question in my mind for a month ago. Mainly I am interested in Hyperbolic Geometry. I found a topic named "Representation Theory of Surface Groups". Let me tell about what is a "...
0
votes
1answer
34 views
Family of stable curves over disc
Let $(C,x)$ be a fixed curve of genus $g \geq 1$ with one marked point and let $D\in C$ be a small disc centered at $x$. Over the punctured disc $D^*$ we have a natural family of smooth curves with ...
2
votes
0answers
36 views
Properness of the coarse moduli map in Keel-Mori theorem.
Given a stack $\mathscr X$ with enough assumptions we obtain a map $\rho: \mathscr X \to X$ to a coarse moduli space. Furthermore, $\rho$ is proper.
I do not understand what it means for $\rho$ to be ...
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0answers
65 views
Universal maps and moduli spaces
I am now reading this wonderful thesis on moduli spaces by
Djounvona.
On Proposition 2.5.3 the author says that if you have a moduli functor $\mathcal{M}$ that is represented by an object $M$ in a ...
4
votes
0answers
53 views
Proof verification : Is my modified moduli problem still representable?
I came across a moduli problem which looks very similar to another one appearing in a little lemma by Rapoport and Zink. I tried to prove that mine is also representable, but as a newbie in this topic ...
1
vote
1answer
61 views
Sketches of stable 6-pointed curves
I was wondering if I could ask about stable $n$-pointed curves (essentially intersecting copies of $\mathbb{P}^1$ with at least $3$ special points on each copy - these may be either a point of ...
3
votes
1answer
82 views
Description of Euler Characteristic Using $\operatorname{Ext}^i$'s
I am reading Nakajima's Lectures on Hilbert Schemes of Points on Surfaces, and I am confused about a detail at the top of page 13. He is sketching a proof that $X^{[n]}(=$ Hilbert scheme of $n$ points ...
2
votes
0answers
43 views
Embedding of moduli space of J-holomorphic spheres into moduli space of stable maps
Let $(M,\omega)$ be a closed symplectic manifold of dimension $2m$, $J$ be an $\omega$-compatible almost complex structure, and $A\in H_2(M;\mathbb{Z})$ a homology class. Let $n \geq 3$ and denote by $...
2
votes
0answers
52 views
Two equivalent definitions of the coLie sheaf for a family of elliptic curves [duplicate]
Let $f: E\rightarrow S$ be a family of elliptic curves, i.e. a proper flat morphism of schemes whose geometric fibers are genus $1$ curves, along with a section $e:S\rightarrow E$. Then there is an ...
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0answers
23 views
Why polarization in Torelli theorem?
Are there two non-isomorphic compact Riemann surfaces with isomorphic integral Hodge structure on $H^1(-,\mathbb{Z})$?
Recall Torelli theorem for Riemann surface: For two compact Riemann surfaces $X,Y$...
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votes
0answers
40 views
Dual Graphs of T-fixed Stable Maps
The following screenshot is taken from "Localization of Virtual Classes" paper of Graber and Pandharipande. I don't understand the following highlighted sentences. From those sentences I ...
0
votes
0answers
45 views
Morphism from Automorphism group of stable curve to automorphism group of its dual graph surjective
I currently study the strata of the moduli stack of algebraic curves and have understood, that for a stable curve with markings $p_1, \dots, p_n$ there is a morphism of groups $Aut(C, p_1, \dots, p_n) ...
2
votes
1answer
75 views
Flabby representable sheaves
Let $S$ be a scheme. Consider some representable moduli functor $\mathcal{M}:(Sch/S)^{op}\rightarrow Set$ represented by some scheme $M$. Then for each $V\in (Sch/S)^{op}$, let define
$$\mathcal{M}^{...
3
votes
1answer
76 views
Moduli Space of Tori
I'm looking at an exercise that reads:
Problem. Let $A = \begin{bmatrix}a & b\\c & d\end{bmatrix} \in GL(2, \mathbb{C})$ and $\Lambda = \langle z \mapsto z + \omega_1, z \mapsto z + \omega_2\...
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vote
0answers
41 views
Satake compactification for the moduli space of curves of genus $g$
I am using the book Compact Complex Surfaces by Wolf P. Barth,
Klaus Hulek, Chris A. M. Peters, Antonius Van de Ven. On p.220, this book says "It follows readily from the projectivity of the ...
2
votes
0answers
37 views
What can we say about moduli spaces of sheaves with double the Chern character of known moduli space of sheaves?
Say I know everything about the moduli space $M_G(v)$ of Gieseker (alternatively, slope, Bridgeland, etc.) stable sheaves with Chern character $v$ on a smooth complex projective variety $X$, where $\...
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0answers
33 views
Why is the Orbifold Euler characteristic of $M_{1, 1}$ equal to $\zeta(-1) $
The orbifold euler characteristic of $M_{1, 1}$= $PSL(2, \mathbb{Z}) $/$\mathbb{H}$is equal to -1/12.
This is due to the fact that $M_{1,1}$ has one two-cell with $\mathbb{Z_{2}}$ automorphism, two ...
2
votes
0answers
127 views
Hilbert scheme of hypersurfaces in Nitsure’s Construction of Hilbert and Quot Schemes
I have a few questions on the construction of Hilbert scheme of hypersurfaces I found recently in Nitin Nitsure’s Construction of Hilbert and Quot Schemes (page 6 part 4)
In his paper Nitsure added a ...
5
votes
2answers
158 views
Relation/Difference between moduli spaces and classifying spaces.
From what I have read so far, a classifying space is a representing object of some (co)representable functor. For example, the $n^\text{th}$ Eilenberg–MacLane space is the classifying space for the $...
9
votes
2answers
294 views
Coarse moduli space of relative Picard functor for affine line
Consider the relative Picard functor $\mathrm{Pic}_{\mathbb A^1/\mathrm{Spec}(\mathbb C)}$ sending a complex scheme $X$ to $\mathrm{Pic}(X \times \mathbb A^1)/\pi_X^* \mathrm{Pic}(X)$.
Since $\mathrm{...
1
vote
0answers
64 views
Singularities of moduli spaces $M_g$
This survey paper from Lizhen Ji says:
Teichmüller was aware that nontrivial automorphisms of Riemann surfaces caused difficulty in constructing $M_g$ and singularities of $M_g$, and he
...
3
votes
0answers
41 views
Morphisms between moduli spaces
Assume that I have two moduli spaces $M_1,M_2$ solving two problems $F_1,F_2:C\to Sets$, think for example about $C=Sch/S$.
Furthermore I have a morphism $f:M_1\to M_2$ or equivalently a natural ...
0
votes
0answers
21 views
Is the product group of finitely many copies of the multiplicative group of positive real numbers a reductive group?
Let G be the multiplicative group of positive real numbers. Is the finite product group $G \times \cdots \times G$ reductive?
I am trying to construct the moduli space for some quiver representations ...
1
vote
1answer
55 views
the moduli spaces that contain direct sums of line bundles
Let $U_X(n,d)$ be the moduli space of semistable vector bundle of rank $n$ and degree $d$ over a smooth projective curves over the complex numbers.
How do I know if $U_X(n,d)$ contains direct sums of ...
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0answers
45 views
When is the coarse moduli space of genus $g$ stable nodal curves singular?
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 ...
2
votes
0answers
28 views
Moduli functor with no coarse moduli space
I have the following problem. Let $\mathcal{M}$ be the set of isomorphism classes of invertible complex 3$\times$3 matrices. If we have a variety $X$, a family over $X$ is a matrix $A(x)$ with ...
2
votes
1answer
821 views
Simple example of a moduli space?
This relates to my previous question: Is $\zeta_{\Bbb R^n}$ a space of moduli spaces?
Moduli Space
I asked two professors about this but they said they didn't know enough about moduli spaces to help....
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0answers
57 views
Dimension of degree $2$ morphisms $\mathbb P^1 \to \mathbb P^1$
I am reading this introduction paper on Gromov-Witten theory, and on page 5, in second paragraph he says:
Within this locus of maps, there is a sublocus consisting of those maps
which map as $2 : ...
2
votes
0answers
77 views
Royal Road to Moduli Spaces
I would like to ask if someone could give me an outline of fundamental works and articles to understand the most important moduli spaces, such as the moduli space of curves, vector bundles, etc... It ...
4
votes
0answers
193 views
On the functorial point of view in algebraic geometry.
Here's a question I've been thinking about lately. I hope it's not too vague - I apologize in advance if this should be the case.
Suppose you want to do algebraic geometry using the $\textit{...
3
votes
0answers
78 views
Birational equivalence of hypersurfaces
Consider all the smooth hypersurfaces of degree $d$ in $\mathbb P^n$, for $d,n$ general enough. If $X$ and $Y$ are birational equivalent, then I think they are not necessarily isomorphic. I want to ...
0
votes
0answers
24 views
Abelian Hitchin's equations
I'm a physicist by trade, and I've recently been working with the dimensional reduction of 4d supersymmetric gauge theories. In particular, the GL-twisted $\mathcal{N}=4$ gauge theory with a simple ...
0
votes
1answer
110 views
References for Higgs bundles
I would like to ask if someone knows good references for the construction of the moduli space of Higgs bundles from the G.I.T point of view, and the study of this moduli space from the stacky point of ...
